Matrix Colorings of P 4 -sparse Graphs

Transcription

1 Diplomabeit Matix Coloings of P 4 -spase Gaphs Chistoph Hannnebaue Januay 23, 2010 Beteue: Pof. D. Winfied Hochstättle FenUnivesität in Hagen Fakultät fü Mathematik und Infomatik

2 Contents Intoduction iii I. Peliminaies 1 1. Types of gaphs Simple Gaphs Cogaphs P 4 -spase Gaphs Genealized Coloings Genealized Coloings with lists II. Main Pat List M-patitions of P 4 -spase Gaphs Peliminay emaks about list M-patitions The list M-patition poblem with fixed-size matices The list M-patition poblem with vaiable-sized M Maximum size of a minimal M-obstuction gaph with lists Minimal M-obstuction, P 4 -spase Gaph without lists Uppe Bounds fo minimal M-obstuction, P 4 -spase Gaphs with bound clique sizes Uppe Bounds fo minimal M-obstuction, P 4 -spase Gaphs Minimal M-obstuction, P 4 -spase Gaphs with constant matices M (,, )-block matices Staicase-like (a, b, c)-block matices Summay 57 Bibliogaphy 58 ii

3 Intoduction A lot of poblems that ae difficult to solve on gaphs in geneal wee poved to be efficiently solvable on specific classes of gaphs. Fo example, gaph isomophism and gaph coloing ae NP-complete in geneal, but can be solved in polynomial time on the class of P 4 -fee gaphs. A pomising paadigm to solve difficult poblems on a class of gaphs involves a unique tee epesentation fo each gaph in the class. Evey leaf in the tee epesents one vetex in the gaph and the inne nodes of the tee epesent opeations with which the gaph fo this node and eventually the gaph fo the whole tee can be constucted fom the gaphs of its subtees. If the class of a gaph as well as its tee epesentation can be detemined in polynomial time, this tee epesentation can be used to ceate polynomial-time algoithms fo poblems that ae had on geneal gaphs. P 4 -fee gaphs ae also known as cogaphs, hence the name cotee fo thei tee epesentation. The class of cogaphs is a subclass of the class of P 4 -spase gaphs that allows only a "low density" of induced P 4 subgaphs. P 4 -spase gaphs also have a unique tee epesentation, which indicates that the poblems efficiently solvable on cogaphs may also be efficiently solvable on P 4 -spase gaphs. Some of the poblems that ae NP-complete fo all gaphs and efficiently solvable fo cogaphs involve a genealized coloing known as matix patition. In the standad coloing poblem, an algoithm has to decide whethe a gaph is k-coloable, that is whethe its vetices can be patitioned into k independent sets. The matix patition poblem allows additional conditions, such as whethe the vetices of a gaph can be patitioned into k independent sets and l cliques and thee can even be estictions on whethe the vetices of a specific pat may o must o must not be adjacent to the vetices of anothe pat. An even futhe genealization of the matix patition poblem is the matix patition poblem with lists. In addition to the conditions of the matix patition, a vetex may only be placed in pats allowed by its list. The tee epesentation of a gaph can be used to find such a matix patition o detemine that no patition exists fo the gaph. A gaph G is an M-obstuction if thee is no patition of the vetices of G such that the coloing conditions defined by the the patition matix M ae satisfied. A minimal M-obstuction is an M-obstuction that contains no M-obstuctions othe than itself as an induced subgaph. The possible size of minimal M-obstuctions has an uppe bound depending on the class of G and M. Specific values fo these uppe bounds can be poved with the help of a tee epesentation. iii

4 Intoduction This thesis is based upon an aticle by Fede, Hell, and Hochstättle [FHH06] featuing a lineatime algoithm to decide the matix patition poblem with lists on cogaphs and that poves uppe bounds fo the size of minimal matix obstuctions cogaphs. The theoems and algoithms can be genealized to also include P 4 -spase gaphs. This genealisation is the main pat of this thesis. Chaptes 1 and 2 define the tems needed in the est of this thesis. A polynomial-time algoithm fo the matix patition poblem with lists fo P 4 -spase gaphs is pesented in Chapte 3, next to a poof that the poblem becomes NP-complete if the matix defining the coloing conditions is pat of the poblem, even fo complete gaphs. The est of this thesis focuses on minimal matix obstuctions. It is poved that the size of a P 4 -spase minimal matix obstuction with lists gows at most factoially. If no lists ae involved, an exponential uppe bound fo the size of minimal matix obstuctions that ae P 4 -spase is shown in Chapte 4. By esticting to constant matices, a polynomial uppe bound can be poved in Chapte 5. iv

5 Pat I. Peliminaies 1

6 1. Types of gaphs Although some gaph poblems ae geneally had to solve, thee ae classes of gaphs allowing these poblems to be solved efficiently. This chapte defines some of these classes and shows how they elate to each othe. Befoe that, some basic conventions ae defined: N shall contain all positive whole numbes including 0, as opposed to N N {0}. Fo m N, let the sets N m N and N m N be defined as N m {x N x m} and N m {x N x m}. When X is a set, its powe set is witten as P(X) {Y Y X}. The symbol stands fo the logical AND, the symbol is a logical OR. The magnitude of a function s slope can be descibed with the big O notation: Let f N R and g N R be functions, then f O(g) lim sup n f (n) g(n) <. This may less fomally be witten as f (n) O(g(n)), whee the paamete name is usually n o m Simple Gaphs All gaphs in this thesis ae simple and the tem gaph is used as a shot fom fo simple gaph. Definition 1. A simple gaph G is a tuple (V, E) with V (the vetices) and E (the edges) being finite sets. The set of edges E is a subset of {{v, w} v, w V, v w}, which is the set of two-element subsets of V. The size G of a gaph G = (V, E) is the numbe of its vetices, G V. Two vetices v 1, v 2 V ae adjacent if {v 1, v 2 } E and non-adjacent if {v 1, v 2 } E. v 1 v 2 {v 1, v 2 } is a shot fom fo an edge. The set of all vetices adjacent to v 1 V is N(v 1 ) {v 2 V v 1 v 2 E}, the set of all vetices non-adjacent to v 1 is N(v 1 ) {v 2 V v 1 v 2 E}. v 1 V distinguishes between v 2 V and v 3 V if v 1 is adjacent to exactly one of the vetices v 2 and v 3 : v 1 v 2 E v 1 v 3 E. Remak 1. Simple gaphs have no paallel edges: Fo evey two adjacent vetices v 1, v 2 V, thee is exactly one edge e E with e = v 1 v2. 2

7 1. Types of gaphs Remak 2. Simple gaphs have no loops: If v 1, v 2 V ae adjacent vetices, then v 1 v 2. Definition 2. The complement gaph G = (V, E) of a gaph G = (V, E) is the gaph with the set of edges E {v 1 v 2 v 1, v 2 V v 1 v 2 E}. That means that two vetices v 1, v 2 V, v 1 v 2 ae adjacent in G if and only if they ae non-adjacent in G. Definition 3. A gaph G = (V, E) containing evey possible edge is called a complete gaph, i.e. E = {v 1 v 2 v 1, v 2 V v 1 v 2 }. The complete gaph with n vetices is K n. The complement K n = (V, ) of a complete gaph is an empty gaph. Definition 4. A gaph G = (V, E ) is called an induced subgaph of G = (V, E) if V V and any two vetices in G ae adjacent if and only if they ae adjacent in G. This may also be witten as G G o G = G V. Similaly, G V is the induced subgaph G (V V ) of G. Definition 5. The set of vetices of a complete induced subgaph is called a clique, theefoe any set of paiwise adjacent vetices is a clique. Similaly, a set of paiwise non-adjacent vetices is called an independent set. Definition 6. A gaph G = (V, E) is k-coloable if thee is a patition A 1,..., A k of its vetex set V = k i=1 A i, consisting of k N paiwise disjoint pats, such that evey pat A i (1 i k) is an independent set. A gaph G = (V, E) is pefect if, fo evey induced subgaph G = (V, E) G, thee is a k N such that the lagest clique of G has size k and G is k-coloable Cogaphs Cogaphs ae a class of simple gaphs that allow a lot of had gaph algoithms to be solved efficiently. In ode to define cogaphs, the gaph P 4, the chodless path of length 3, will be defined fist: Definition 7. The chodless path of length 3, P 4, is a simple gaph (V, E) with fou vetices {v 1, v 2, v 3, v 4 } = V and exactly thee edges {v 1 v 2, v 2 v 3, v 3 v 4 } = E. Remak 3. The complement gaph of P 4 is P 4. In the 1970s, vaious eseaches discoveed cogaphs independently, although they used diffeent definitions and names. Jung used the tem D -gaph [Jun78], Seinsche woked on them as gaphs that have no induced subgaph isomophic to P 4 [Sei74], and Sumne called them HD (o Heeditay Dacey) gaphs [Sum74]. Definition 8. A cogaph is a gaph (V, E), which does not have a P 4 as an induced subgaph. This means, that fo any fou vetices v 1, v 2, v 3, v 4 V with v 1 v 2, v 2 v 3, v 3 v 4 E, thee must be anothe edge e E {v 1 v 3, v 1 v 4, v 2 v 4 }. 3

8 1. Types of gaphs Thee ae a couple of othe chaacteizations of cogaphs. The most impotant chaacteization is the cotee epesentation of a cogaph. [CLB81] Definition 9. A cotee T(G) = (V T(G), E T(G) ) is a tee epesenting a gaph G = (V, E). T(G) s inne nodes ae labeled o + and its leaves ae exactly the vetices of G. Two vetices of G ae connected if and only if thei least common ancesto in T(G) is labeled +. A nomalized cotee will have its inne nodes labeled + and in alteation, which means that a node s paent will always be labeled + and the othe way aound. All gaphs epesented by a cotee can also be epesented by a nomalized cotee since a node and its child having the same label can be contacted to one node. This will obviously not change the label of the least common ancesto fo any two leaves. Two nomalized cotees epesenting the same gaph will be isomophic. Nomalized cotees ae theefoe unique epesentations of cotee-epesentable gaphs. A binay cotee is a cotee in binay fom, that means evey node has exactly two child nodes o it is a leaf. This cotee epesentation is also unique if a pope algoithm fo splitting nodes with moe than two child nodes is used fo its constuction. Figues 1.1 and 1.2 show a cogaph and its binay cotee epesentation. Figue 1.1.: An example of a cogaph Figue 1.2.: Binay cotee coesponding to the cogaph Coneil et al. have poved that a gaph is a cogaph if and only if it has a cotee epesentation [CLB81]. A gaph can be built fom its cotee in polynomial time by assigning each node t V T(G) a gaph G t. Fo each leaf v V, the gaph G v is the single vetex gaph G v = ({v}, ) containing only the leaf vetex itself. Fo an inne node t V T(G) V labeled, the gaph G t is the disjoint union G u1... G un of the gaphs of its n child nodes u 1,..., u n in T(G). Fo an inne node t V T(G) V labeled +, the gaph G t is the join G u G un (V u1... V un, E u1... E un {x y 1 i, j n i j x V ui y V u j } of the gaphs of t s child nodes u 1,..., u n in T(G). The gaph G assigned to the oot V T(G) of T(G) is the cogaph that T(G) epesents. Coneil et al. have poven that the unique cotee of a cogaph can be computed in linea time [CPS85], 4

9 1. Types of gaphs Betsche et al. have developed an easie cogaph ecognition algoithm that uns in linea time [BCHP08] P 4 -spase Gaphs Anothe class of simple gaphs ae the P 4 -spase gaphs. P 4 -spase gaphs constitute a supeclass of cogaphs. Hoàng fist defined this type of gaph and showed besides othe things that P 4 - spase gaphs ae pefect [Hoà85]. Definition 10. A simple gaph G = (V, E) is called a P 4 -spase gaph if evey subgaph of G induced by any five vetices in V contains at most one P 4 as an induced subgaph. Extending the available opeations in the cogaph s cotee epesentation by additional node opeations yields a unique tee epesentation of P 4 -spase gaphs. Jamison and Olaiu have shown how to acquie this tee epesentation in polynomial time [JO92a]. An impotant definition fo this tee epesentation is the gaph class called a spide: Definition 11. A simple gaph G = (V, E) is called a spide if its vetex set splits into an even numbe 2ν 4 of vetices c 1,..., c ν, s 1,..., s ν and a vetex set R, such that c 1,..., c ν compise a clique. Evey two vetices c i, c j (1 i, j ν) ae adjacent. s 1,..., s ν compise an independent set. Evey two vetices s i, s j (1 i, j ν) ae nonadjacent to each othe. Each vetex c i (1 i ν) is adjacent to all vetices in R. No vetex s i (1 i ν) is adjacent to a vetex in R. Eithe the vetices c i and s j ae adjacent if and only if i = j (slim spide), o the vetices c i and s j ae adjacent if and only if i j (fat spide). The vetices c 1,..., c ν ae the body of the spide, the vetices s 1,..., s ν ae the spide s legs, and the set R is the spide s head. If R is an empty set, the gaph is called a headless spide. An example of a spide gaph is shown in Figue 1.3. The class of spide gaphs is the basis fo two new gaph opeations. Definition 12. Let s 1,..., s ν, c 1,..., c ν be vetices with ν 2. Let R = (V R, E R ) be a gaph. Let E CR {c i 1 i ν, V R }. 5

10 1. Types of gaphs Figue 1.3.: Example of a slim spide gaph with ν = 8 and R = { 1, 2, 3 } The functions and ae defined as c 1,..., c ν, s 1,..., s ν, R ({c 1,..., c ν, s 1,..., s ν } V R, E slim E R E CR ) c 1,..., c ν, s 1,..., s ν ({c 1,..., c ν, s 1,..., s ν }, E slim ) c 1,..., c ν, s 1,..., s ν, R ({c 1,..., c ν, s 1,..., s ν } V R, E fat E R E CR ) c 1,..., c ν, s 1,..., s ν ({c 1,..., c ν, s 1,..., s ν }, E fat ) whee E slim {c i s i 1 i ν} and E fat {c i s j 1 i, j ν i j}. Obviously, maps to slim spide gaphs and maps to fat spide gaphs. The P 4 -spase gaph s tee epesentation extends cotees by the two labels and fo inne nodes. When constucting the gaph G = (V, E) epesented by the tee T(G) = (V T(G), E T(G) ), evey node t V T(G) is assigned a gaph G t. Again, the tee s leaves ae the gaph G s vetices. An inne node t V T(G) now has fou possible opeations to ceate its gaph out of its childen, depending on its label: A node labeled is assigned the gaph ceated by the disjoint union of the gaphs of its child nodes, which is the same as fo cotees. A node labeled + is assigned the join of the gaphs of its child nodes, again identical to cotees The gaph G t (c 1,..., c ν, s 1,..., s ν,g ) of a node labeled is the opeation applied to its leaf child nodes c 1,..., c ν, s 1,..., s ν and the gaph of its child. If a node labeled has an even numbe of leaf child nodes c 1,..., c ν, s 1,..., s ν and no futhe childen, then G t (c 1,..., c ν, s 1,..., s ν ) is a slim headless spide gaph. 6

11 1. Types of gaphs Similaly, the gaph G t (c 1,..., c ν, s 1,..., s ν,g ) of a node labeled is the opeation applied to its child leaves c 1,..., c ν, s 1,..., s ν and the gaph of the its child node. If a node labeled has an even numbe of leaf child nodes, c 1,..., c ν, s 1,..., s ν, then G t (c 1,..., c ν, s 1,..., s ν ) is a fat headless spide gaph. This definition of a tee epesentation fo P 4 -spase gaph diffes a little fom the oiginal tee epesentation defined by Jamison and Olaiu, but thei poof that a gaph is P 4 -spase if and only if it can be epesented by such a tee applies hee as well [JO92a]. They have also shown in a late publication that the tee epesentation of a P 4 -spase gaph can be computed in linea time [JO92b]. Figue 1.4 shows an example of a P 4 -spase gaph next to its tee epesentation in Figue 1.5. Figue 1.4.: An example of a P 4 -spase gaph Figue 1.5.: Tee epesentation coesponding to the P 4 -spase gaph Definition 13. A split gaph G = (C S, E) consists of a clique C and an independent set S. G is called avoiding if fo all s S and all c C, the numbe of vetices in C non-adjacent to s is N(s) C 2 and the numbe of vetices in S adjacent to c is N(c) S 2. G is called minimal avoiding if G has the following popeties: 1. Evey vetex c C is non-adjacent to a vetex s S such that N(s) C = Evey vetex s S is adjacent to a vetex c C such that N(c) S = G is avoiding. Remak 4. Let the split gaph G = (C S, E) be avoiding. Then the vetices of G split into a clique C and an independent set S with C, S, and C S =. Poposition 1. Let G = (C S, E) be an avoiding split gaph. Then G contains a minimal avoiding gaph as an induced subgaph. Poof. This poof will show that evey avoiding split gaph that does not aleady satisfy the popeties of a minimal avoiding split gaph contains a vetex x C S such that G {x} is an avoiding split gaph. This way, vetices can be emoved fom the gaph until we get a minimal avoiding induced subgaph of G. As G contains only a finite numbe of vetices, such an induced subgaph can always be found. 7

12 1. Types of gaphs Thus, assume that fo the avoiding split gaph G does not satisfy at least one of the fist two popeties of minimal avoiding gaphs. By complementing G if necessay, we may assume that popety 1 is not satisfied, hence thee is a vetex c C fo which all vetices s N(s) have N(s) C 3. Any induced subgaph of a split gaph obviously is a split gaph and G {c} also is avoiding as the condition fo avoiding gaphs is affected only fo vetices in S and of couse only those that have been non-adjacent to c befoe its emoval; since these vetices s N(c) had the popety N(s) C 3 in G, they still satisfy the avoiding condition N(s) (C {c}) 2 in G {c}. Poposition 2. An avoiding split gaph G = (V, E) is not P 4 -spase. Poof. Assume that the poposition is false and let G = (V, E) be a counteexample with a minimum numbe of vetices. By Poposition 1, G has a minimal avoiding subgaph and, as G has a minimum numbe of vetices, G itself must be minimal avoiding. Since G is a split gaph, thee is a vetex s 1 S. G is minimal avoiding, so s 1 is adjacent to a vetex c 1 C with N(c 1 ) S = {s 1, s 2 }. Let s 3 S be a vetex non-adjacent to c 1 with N(s 3 ) C = {c 1, c 2 }. Let c 3 C be a vetex adjacent to s 3 such that c 3 is adjacent to exactly two vetices in S. The gaph G {s 1, s 2, s 3, c 1, c 2, c 3 } that was constucted so fa is depicted in Figue 1.6 with a legend to be found in Figue If c 3 is adjacent neithe to s 1 no s 2, then G {s 1, s 2, s 3, c 1, c 3 } is a gaph of size 5 with two induced P 4 s, contadicting that G is P 4 -spase. This contadiction is shown in Figue 1.7. Thus, we may assume that c 3 is adjacent to s 2 and non-adjacent to s 1. If c 2 is adjacent to s 1, then G {s 1, s 3, c 1, c 2, c 3 } would be a gaph of size 5 with two induced P 4 s, which is shown in Figue 1.8. As G is avoiding, c 2 is adjacent to a vetex s 4 S, s 4 s 2. Because s 4 {s 1, s 2 } = N(c 1 ) S, we see c 1 s 4 E. Anothe consequence is c 2 s 2 E as othewise G {s 1, s 2, s 4, c 1, c 2 } would be a gaph of size 5 with two induced P 4 s. Figue 1.9 shows the contadiction if c 2 s 2 E. G is avoiding, so thee ae two vetices c 4, c 5 C, c 4 c 5 that ae non-adjacent to s 2. Since {c 4, c 5 } {c 1, c 2 } = and N(s 3 ) C = {c 1, c 2 }, we conclude that s 3 is adjacent to c 4 and c 5. G {s 2, s 3, c 1, c 4, c 5 } yields the final contadiction, as shown in Figue Theoem 1. Evey P 4 -spase gaph with vetices contains a maximum clique C and a maximum independent set S that meet in one vetex, C S = 1. Poof. Let G = (V, E) be a P 4 -spase gaph with C V being a maximum clique and S V being a maximum independent set. We may assume C S = as a clique and an independent set may intesect only in at most one vetex and in that case we would aleady have C S = 1. G is not avoiding, so thee is a vetex s S with N(s) C < 2 o a vetex c C with N(c) S < 2. By complementing G if necessay, we may assume s S with N(s) C 1. Actually we have N(s) C = 1 since N(s) C = 0 implies that C {s} would be a lage clique than C. Thus, 8

13 1. Types of gaphs Figue 1.6.: Six vetices of G Figue 1.7.: The subgaph is not P 4 -spase Figue 1.8.: Again, G is not P 4 -spase Figue 1.9.: Next subgaph that is not P 4 - spase Figue 1.10.: The final contadiction, G is not P 4 -spase Figue 1.11.: Legend fo the othe figues thee is exactly one vetex c s C that is non-adjacent to s. As a esult, C {c s } {s} induces a maximum clique in G that meets the maximum independent set S in exactly one vetex, s. 9

14 2. Genealized Coloings The oiginal n-coloing poblem asked whethe the vetices of a gaph can be patitioned into n independent sets. Although thee ae diffeent genealizations fo this poblem, this thesis focuses on matix patitions, which does not only add pats of cliques, but can also place estictions on the edges between vetices of diffeent pats. Theefoe, seveal well-known gaph patition poblems can be epesented as a matix patition, which was fist defined by Fede et al. [FHKM99]. Definition 14. Let M {0, 1, } m m be a symmetic matix with m ows and m columns with enties eithe 0, 1, o. Then M is called a patition matix. The gaph G = (V, E) admits M if its vetices can be patitioned into m disjoint sets A 1,..., A m such that fo evey i, j N m, i j, if M i,i = 0, then A i is an independent set, if M i,i = 1, then A i is a clique, if M i,i =, then thee is no estiction on the edges between vetices of A i, if M i, j = 0, then evey vetex in A i is non-adjacent to evey vetex in A j, if M i, j = 1, then evey vetex in A i is adjacent to evey vetex in A j, and if M i, j =, then thee is no estiction on the edges between A i and A j. The patition A 1,..., A m is called an M-patition. The numbes 1,..., m uniquely identify a set of vetices in the patition A 1,..., A m. Theefoe these numbe may also be called pats in ode to highlight thei meaning in this context. If a gaph G does not admit the matix M, then G obstucts M. G is a minimal M-obstuction if G obstucts M and all induced subgaphs of G except G itself admit M. A patition submatix M of a matix M is a submatix of M such that the diagonal enties in M ae diagonal enties in M. Definition 15. Let M m µ=1 {0, 1, }µ µ be a set of symmetic matices and let G be a gaph. G admits M if thee is a matix M M that G admits. If G obstucts all matices M M then G obstucts M. G is a minimal M-obstuction if G obstucts M and all induced subgaphs of G except G itself admit M. 10

15 2. Genealized Coloings The decision whethe a gaph admits a matix M o not is the M-patition poblem. The geneal matix patition poblem may also be called M-patition poblem, even if M is not explicitly defined. Some poblems equie the sets A i (i N m) to be non-empty, which will not be the focus of this thesis, although some esults may also apply to these poblems. Definition 16. Let M {0, 1, } m m be a symmetic matix. Then M {0, 1, } m m denotes the complement of M and, fo all i, j N m, the enties of M satisfy the conditions M i, j = 0 M i, j = 1 and M i, j = 1 M i, j = 0. Definition 17. Let M {0, 1, } m m be a symmetic matix and P, Q N m. The matix M P,Q {0, 1, } P Q is defined as the submatix of M whee only the ows in P and only the columns in Q ae taken. M P {0, 1, } P P is a shot fom fo the patition submatix M P M P,P Genealized Coloings with lists Fede et al. [FHKM03] and Cameon et al. [CEHS04] applied the list concept [AT92, FS92] to the matix patition poblem as a futhe genealization. Hee, the vetices cannot be placed into any abitay pat, instead thee is a pedefined list of pats fo evey vetex detemining to which pats the vetex may belong: Definition 18. Let the matix M {0, 1, } m m with m ows and m columns be symmetic and each of its enties is eithe 0, 1, o. Let thee be a gaph G = (V, E). The function L V P (N m) assigns a set of pats to each vetex in G. Each set L (v) N m is called a list, while the function L is efeed to as lists. The gaph G with lists L admits M if thee is an M-patition A 1,..., A m such that v A i i L (v) (v V, 1 i m) In this case, A 1,..., A m is called a list M-patition (o just M-patition if it is clea fom the context that lists ae involved). The matix patition poblem without lists is identical to the matix patition poblem with lists if all lists L (v) ae set to N m. With these lists, thee ae no estictions on whee the vetices can be placed. Theefoe, all algoithms fo list matix patition poblems can also be used fo matix patition poblems without lists. Remak 5. A gaph G with lists L has an M-patition A 1,..., A m if and only if A 1,..., A m is an M-patition of G with lists L. 11

16 2. Genealized Coloings Definition 19. Let the matix M {0, 1, } m m be symmetic and let G = (V, E) be a gaph with lists L. Let G = (V, E ) be an induced subgaph of G. Then the lists L V V P (N m) ae defined as L V (v ) L(v ) fo all v V. The fom L G L V may also be used fo the same function. With this definition, list M-patitions fo subgaphs can be descibed by esticting the list to the subgaph s vetex subset: Definition 20. Let the matix M {0, 1, } m m be symmetic and let G be a gaph with lists L. G is a minimal M-obstuction with lists L if G obstucts M with lists L and evey induced subgaph G of G admits M with lists L G. If patition submatices of M ae involved, evey list s pats have to be changed such that the numbes in the list still efes to the same ows and columns in the matix. This is accomplished with the help of a mapping function: Definition 21. Let m N, P N m, then η P P N P is defined as η P x P N x, i.e. x is the η P (x)th element of P egading the natual ode of N). Its invese function η 1 P N P P theefoe is η 1 P (y) = min {x N y = N x P }, i. e. η 1 P (y) is the yth element of P. Let M P {0, 1, } P P be a patition submatix of M {0, 1, } m m and let L V P (N m) be lists fo M. Then the lists P L V P (N P ) ae defined as PL(v) η P (P L(v)) = {η P (x) x P L(v)}, i.e. the pats in P ae enumeated as 1,..., P in the lists while pats not in P ae discaded. If M is a patition submatix of M, the elationship between list M -patitions and list M- patitions can be descibed with these estictions of L: Definition 22. Let the matix M {0, 1, } m m be symmetic and let M m µ=1 {0, 1, }µ µ be a collection of patition submatices of M. Let G be a gaph with lists L V P (N m). Then G is an M-obstuction with lists L if, fo evey matix M M and evey P N m with M P = M, G obstucts M with lists P L. G is a minimal M-obstuction with lists L if G is an M-obstuction with lists L and no induced subgaph G of G othe than G itself is an M-obstuction with lists L G. In this thesis, a set of matices M and lists L may be given without fomally mentioning which matix M the matices in M ae patition submatices of and fo which matix the lists L ae defined. This applies especially to the following Chapte 3, which discusses list M-patitions. In these cases, the matices in the set ae patition submatices of the matix with the symbol M, which is defined as a symmetic matix M {0, 1, } m m thoughout the majo pat of the chapte. 12

17 Pat II. Main Pat 13

18 3. List M-patitions of P 4 -spase Gaphs 3.1. Peliminay emaks about list M-patitions Theoems vey simila to the following two lemmas have aleady been poved by Fede et al. ([FHH06], Lemma 2.1 and Coollay 2.2), the following poofs ae slightly adapted fo the lemmas found hee. Lemma 1. Let G = G 1 G 2 be a disconnected gaph and M {0, 1, } m m. G admits M with lists L if and only if thee ae sets of pats P, Q N m such that G 1 admits M P with lists P L G1, G 2 admits M Q with lists Q L G2, and M P,Q contains no 1. Poof. Let 1 A i (i P) be an M P -patition of G 1 with lists P L G1 and let 2 A i (i Q) be an M Q - patition of G 2 with lists Q L G2 and let A i 1 A i 2 A i (1 i m) be the join of these patitions, whee 1 A i and 2 A i ae empty sets fo i P and i Q, espectively. The patition A 1,..., A m of G is compatible with the lists L, because fo evey j {1, 2}, i N m, and evey v j A i, we have η P (i) P L G j (v) and thus i L(v). Hence, we just have to validate that the M-patition condition is not violated. Assume the vetices v, w V violate the M-patition condition of A 1,..., A m. Then the vetices v, w will not both be within the same gaph G 1 o G 2, because then they would also violate the patition condition of 1 A i (i P) o 2 A i (i Q). Theefoe, one of the vetices is in G 1 and the othe is in G 2 and theefoe they ae not adjacent. As v, w violate the patition condition, the coesponding matix enty must be M j,k = 1 fo v A j, w A k. On the othe hand, M P,Q does not contain a 1 and this also means M j,k 1, a contadiction. Let A 1,..., A m be an M-patition of G with lists L. Define j A i A i V j fo G j = (V j, E j ), j {1, 2} and i {1,..., m}. Let P, Q be defined via i P 1 A i and i Q 2 A i. Then, obviously, 1 A i (i P) is a valid M P -patition with lists P L G1 and 2 A i (i Q) is a valid M Q -patition with lists Q L G2, so G 1 admits M P with lists P L G1 and G 2 admits M Q with lists QL G2. Assume that M P,Q contains a 1. Then thee ae j P, k Q with M j,k = 1 and, by the definition of P and Q, thee ae vetices v A j V 1 and w A k V 2. These vetices ae non-adjacent, because v is in G 1 and w in G 2 and the two subgaphs ae disconnected. Theefoe A 1,..., A m is no valid patition of G, which contadicts the assumptions. Theefoe M P,Q contains no 1. 14

19 3. List M-patitions of P 4 -spase Gaphs Coollay 1. Let G = G 1 G 2 be a disconnected gaph and M {0, 1, } m m. G obstucts M with lists L if and only if fo all sets of pats P, Q N m the subgaph G 1 obstucts M P with lists P L G1, G 2 obstucts M Q with lists Q L G2, o M P,Q contains a 1. Lemma 2. Let M m µ=1 {0, 1, }µ µ be a set of patition submatices of M {0, 1, } m m and let G = G 1 G 2 be a disconnected gaph with lists L. Fo i {1, 2}, let the matix set M i m µ=1 {0, 1, }µ µ only contain patition submatices of matices in M and let G i be an M i -obstuction with lists L Gi. Fo evey matix M M and all sets of pats P, Q N m, let thee be a 1 in M P,Q, o M P M 1, o M Q M 2. If G is a minimal M-obstuction with lists L, then G 1 is a minimal M 1 -obstuction with lists L G1 and G 2 is a minimal M 2 -obstuction with lists L G2. Poof. Assume G 1 is not a minimal M 1 -obstuction with lists L G1, then G 1 has an induced subgaph G 1 G 1 that obstucts M 1 with lists L G 1. By Coollay 1, the induced subgaph G = G 1 G 2 G of G is an M-obstuction with lists L G, which contadicts that G is a minimal M-obstuction with lists L. Thus, the initial assumption is wong and G 1 is a minimal M 1 - obstuction with lists L G1. Swapping G 1 and G 2 as well as M 1 and M 2 in the poof above shows that G 2 is a minimal M 2 -obstuction with lists L G2. If no lists ae used, the condition that the matices in M ae patition submatices of a matix M {0, 1, } m m may be omitted, as it is equied only to have a pope definition fo the lists of the submatices; a matix M {0, 1, } m m with lists L(v) N m (v V) can always be found by putting all matices in M in the diagonal of M (m is the sum of all matix sizes in M in this case): Coollay 2. Let M m µ=1 {0, 1, }µ µ be a set of matices and let G = G 1 G 2 be a disconnected gaph. Fo i {1, 2}, let the matix set M i m µ=1 {0, 1, }µ µ only contain patition submatices of matices in M and let G i be an M i -obstuction. Fo evey matix M M and all sets of pats P, Q N m, let thee be a 1 in M P,Q, o M P M 1, o M Q M 2. If G is a minimal M-obstuction, then G 1 is a minimal M 1 -obstuction and G 2 is a minimal M 2 -obstuction. As a esult of Coollay 1, matix sets M 1, M 2 as equied by Lemma 2 can always be chosen: fo example, M i could contain all patition submatices M of all matices in M such that G i obstucts M with lists L Gi. Thus, if G = G 1 G 2 is a minimal M-obstuction with lists fo some set M of patition submatices of a matix M, then G 1 is a minimal M 1 -obstuction with lists fo some matix set M 1. The following lemma limits the numbe of vetices that cetain spide gaphs have. This ensues that the time complexity fo some algoithms (namely the ones in Section 3.2 and Section 4.2) on P 4 -spase gaphs pesented in this thesis does not depend on the numbe of vetices n but 15

20 3. List M-patitions of P 4 -spase Gaphs instead on the numbe of patitions m. Fo a fixed matix size m, these algoithms can be shown to become polynomial in n. Definition 23. Let G = (V, E) be a gaph. Two pais of vetices (a, b), (a, b ) V V ae called a twin couple if a b E a b E, and a c E a c E, and b c E b c E (c V {a, b, a, b }). Lemma 3. Let M {0, 1, } m m denote a patition matix and G = (V, E) a gaph with lists L. Let the pais of vetices (a 1, b 1 ),..., (a m+2, b m+s ) V V be paiwise disjoint and twin couples such that L(a i ) = L(a j ), L(b i ) = L(b j ) (1 i, j m + 2). Then G {a 1, b 1 } with lists L V {a1,b 1 } admits M if and only if G admits M with lists L. Poof. If G admits an M-patition with lists L, then G {a 1, b 1 } also admits an M-patition with lists L V {a1,b 1 }. Assume G {a 1, b 1 } has an M-patition with lists L V {a1,b 1 } with pats A 1,..., A m. The m+1 vetices a 2,..., a m+2 ae elements of the m sets A 1,..., A m, so by the pigeon hole pinciple, thee must be a pat with at least two diffeent vetices fom {a 2,..., a m+2 }. Without loss of geneality, we may assume a 2, a 3 A 1. Adding a 1 to A 1 and b 1 to the pat A j containing b 2 yields an M-patition of G, which will now be poved. Fist, we obseve 1 L(a 2 ) = L(a 1 ) and j L(b 2 ) = L(b 1 ), so ou choice of classes fo a 1 and b 1 is compatible with the lists. Now assume that thee exists a pai of vetices x, y violating the patition condition. Since thee is no violation without a 1 and b 1, one of the vetices x, y must be a 1 o b 1. Without loss of geneality, the two cases x = a 1 and y = b 1 may be looked at sepaately. In the case x = a 1, the twin couple popety a 1 c E a 2 c E fo all c V {a 1, b 1, a 2, b 2 } implies y {b 1, a 2, b 2 }. Because (a 1, b 1 ) and (a 3, b 3 ) ae also twin couples, we obseve y {b 1, a 3, b 3 }, which means y = b 1. The case y = b 1 similaly leads to x = a 1 : Because (a 1, b 1 ), (a 2, b 2 ), and (a 3, b 3 ) ae paiwise twin couples, we have b 1 c E b 2 c E (c V {a 1, b 1, a 2, b 2 }) and b 1 c E b 3 c E (c V {a 1, b 1, a 3, b 3 }) and theefoe x {a 1, a 2, b 2 } {a 1, a 3, b 3 }. This means that the two cases ae in fact only one and the pai of vetices (x, y) = (a 1, b 1 ) ae the only vetices violating the patition condition. The vetex pais (a 1, b 1 ) and (a 2, b 2 ) ae a twin couple, hence a 1 b 1 E a 2 b 2 E. Since a 1, a 2 A 1 and b 1, b 2 A j, (a 2, b 2 ) also violates the patition condition, which contadicts the choice of A 1,..., A m as classes fo an M-patition of G {a 1, b 1 }. Hence, the oiginal assumption must be wong and G admits an M-patition. 16

21 3. List M-patitions of P 4 -spase Gaphs 3.2. The list M-patition poblem with fixed-size matices The list M-patition poblem is NP-complete because it is a genealization of the NP-complete coloing-poblem (see Section 2.1). If the matix size m is fixed, the M-patition poblem is still NP-complete. Fo some matices, the M-patition poblem is NP-complete, even if the whole matix is fixed, because fo example Plana 3-coloability educes to an M-patition with a fixed matix [Sto73]. If the gaph is a cogaph and the size m N of the matix M {0, 1, } m m is fixed, thee is an algoithm that decides the list M-patition poblem in linea time [FHH06]. This section genealizes this algoithm to P 4 -spase gaphs. The algoithm solves the list M-patition poblem with a fixed size m of the matix fo P 4 -spase gaphs in linea time. Poposition 3. The list M-patition poblem fo headless spide gaphs can be solved in time O (nm + m 2 16 m ), linea in n. Poof. A headless spide gaph G = ({c 1,..., c ν, s 1,..., s ν }, E) with lists L satisfies the conditions in Lemma 3 if p m + 2 vetex pais (c 1, s 1 ),..., (c p, s p ) have the same lists. These pais of vetices ae twin couples, so emoving the vetices c m+2,..., c p and s m+2,..., s p yields, by Lemma 3, an induced subgaph of G that admits exactly the same matices of size m with coesponding lists. Removing these edundant vetices fo all diffeent list pais leads to the induced subgaph G = (V, E ) of G, which contains at most m+1 vetex pais with identical list pais fo each list pai. G with lists L L G still admits the same matices with size m as G with lists L. As thee ae (2 m ) 2 = 2 2m diffeent combinations of two lists and G contains at most m+1 vetex pais fo each combination, the numbe of vetices in G is n V 2 (m+1) 2 2m. Obviously, this uppe limit fo n does not depend on the numbe n of vetices in G, although calculating G fom G takes linea time in n. Theefoe, checking all patitions whethe they constitute a valid M-patition does not incease the time in egads to n: Fo evey patition to check, evey vetex is an element of one of the m classes. This means that thee ae n m possibilities to distibute the vetices to m classes. Fist, each patition is checked whethe it is compatible with the lists L, which takes the time O (n m) pe patition. Patitions violating the patition condition given by the matix M can be dopped in time O (n 2 ) by compaing the adjacency of evey pai of vetices in pats i, j {1,..., m} to the value of M i, j. All othe patitions ae valid, so G with lists L and theeby G with lists L admits M if and only if at least one valid patition is left. The steps of this algoithm taking the most time ae the calculation of G fom G, which takes the time O(n m), and the adjacency compaison, which takes the time O (n 2 ). As a whole, the algoithm theefoe takes the time O (mn + n 2 ) O (nm + 4m 2 (2 2m ) 2 ) = O (nm + m 2 16 m ). 17

22 3. List M-patitions of P 4 -spase Gaphs The algoithm descibed in this section exploits the tee epesentation of spide gaphs. The matices admitted by the gaph of a node in the tee will be calculated fom the node s childen. The next lemma is a peequisite fo such an opeation on spide nodes. Lemma 4. Let G = (S R, E) be a spide gaph with lists L. The vetices R ae the head of the spide and the est of the vetices S {c 1,..., c ν, s 1,..., s ν } split into the spide s body {c 1,..., c ν } and legs {s 1,..., s ν }. Then G admits a matix M {0, 1, } m m if and only if thee ae submatices M S, M R of M (with S, R N m) such that G R with lists L R admits M R and G S admits M S with lists L defined as follows: L (c i ) L(c i ) {x N m y R.M x,y = 0} L (s i ) L(s i ) {x N m y R.M x,y = 1} Poof. Let the gaph G admit a matix M {0, 1, } m m. Then G has an M-patition with classes A 1,..., A m. Let S N m denote the set of indices i of classes A i containing vetices of S. Similaly, let R N m be the set of indices j of pats A j containing vetices of R. Since G with lists L has an M-patition with classes A 1,..., A m, the same M-patition will be valid if it is esticted to the vetices of the induced subgaph G R with lists L R. Resticting the M-patition to the induced subgaph G S with lists L S also yields a valid patition fo the induced subgaph. As all vetices of R ae in pats A i with i R and all vetices of S ae in pats A i with i S, G R with lists L R will also admit M R and G S with lists L S will also admit M S. The last thing to pove is that G S still admits M S if the lists L instead of L S ae used. By the definition of L, this claim is tue if thee ae no pats x S and y R with a vetex c i A x and M x,y = 0 o a vetex s i A x and M x,y = 1. The fist case is not possible, because y R implies A y R and theefoe c i is adjacent to a vetex A y R. Then M x,y = 0 contadicts that A 1,..., A m constitute an M-patition fo G. The latte case is not possible eithe, because the vetex A y R is non-adjacent to s i, which contadicts M x,y = 1 and A 1,..., A m being a valid M-patition fo G. Hence, G S with lists L admits M S. Let thee be sets S, R N m, such that G R with lists L R admits M R and G S with lists L admits M S. Then G R has an M R -patition R A i (i R) with lists R L R and G S has an M S -patition S A i (i S) with lists L. The est of the sets ae teated as empty sets, R A i (i N m R) and S A i (i N m S). The join A i R A i S A i (1 i m) of these classes constitutes an M-patition fo G with lists L: 18

23 3. List M-patitions of P 4 -spase Gaphs The sets A 1,..., A m ae supesets of the sets R A 1,..., R A m and S A 1,..., S A m and the lists L ae sticte than L, so any two vetices both within the same induced subgaph G R o G S will not violate the patition condition fo A 1,..., A m. Theefoe, if the vetex pai, q violates the patition condition, thee will be one vetex R A y with some y R and one vetex q S A x with some x S. If q = c i fo an i {1,..., ν}, then x L (q) = L (c i ) implies M x,y 0 by the definition of L. This cannot be a violation of the patition condition, because c i and ae adjacent, so we may assume q = s i fo an i {1,..., ν}. Then x L (s i ) = L (q) implies M x,y 1, which is also no violation of the patition condition, because s i and ae not adjacent. Theefoe the patition condition is not violated and the classes A 1,..., A m constitute a patition of G with lists L. Thus, G with lists L admits M. Lemma 4 equies a given patition submatix M S fo the spide body and legs in ode to detemine whethe G admits the matix M. If no such submatix is given, the algoithm fom Poposition 3 efficiently calculates all possible patition submatices. With this peequisites, the matices admitted by a given spide gaph can be calculated in linea time: Poposition 4. Let the gaph G = (S R, E) with lists L be a spide gaph, whee S is the spide s body and legs and R is the spide s head. Let 2ν S be the numbe of vetices in S and let M {0, 1, } m m be a matix. Let the set of matices M R contain all patition submatices of M that the induced subgaph G R of G with lists L R admits. Given G = (S R, E), L, M, and M R, the set of patition submatices of M admitted by G can be calculated in time O (ν m 4 m + m 2 64 m ), linea in ν. Poof. Fo each set of pats R N m such that G R with lists L R admits the submatix M R of M, thee steps ae pocessed. Note that M R M R. Step 1: Let the lists L fo the gaph G S be defined like the lists L in Lemma 4. Calculating L as follows takes the time O (ν m 2 ): Thee ae 2 ν lists to calculate and each of them contains at most m pats. Fo each pat i {1,..., m}, the R m enties fom the ow M i,1,..., M i,m of the matix M have to be checked in ode to decide whethe i lies within the list o not. Step 2: The set M S of all matices that G S admits with lists L can be calculated in time O (2 m ( νm + m 2 16 m )) = O (ν m 2 m + m 2 32 m ) by using the algoithm in Poposition 3 on evey patition submatix of M, of which 2 m exist. Step 3: Fo each set of pats S such that M S M S, Lemma 4 implies that G with lists R S L admits the matix M R S. This takes the time O (m 2 m ), as fo each of the at most 2 m patition submatices in M S, only the at most m pats have to be joined fom the two sets R and S. Step 1 to 3 as a whole need O (ν m 2 m + m 2 32 m ) time, as step 2 dominates the time equiements. With Step 1 to 3 epeated fo each of the 2 m possible sets of pats R N m, all patition 19

24 3. List M-patitions of P 4 -spase Gaphs submatices admitted by G with lists L ae calculated in time O (2 m (ν m 2 m + m 2 32 m )) = O (ν m 4 m + m 2 64 m ) Theoem 2. The M-patition poblem with lists can be decided in linea time fo P 4 -spase gaphs if the size of M is constant and the gaph is given in its tee epesentation. Poof. Let G = (V, E) be a P 4 -spase gaph with lists L. Fo each vetex set V V, let the matix set M V M be the set of patition submatices admitted by G V with lists L V. Each node t of the tee epesentation of G is assigned a vetex set V t V, such that the subtee ooted at t epesents the induced subgaph G V t. The set M Vt containing all patition submatices of M admitted by the induced subgaph G V t with lists L Vt will be calculated fo each node t fom the leaves to the oot: A leaf node maps to exactly one vetex v V. Fo any patition P N m, the leaf s gaph G {v} with lists P L {v} admits M P if and only if L(v) P. Calculating M {v} theefoe takes the time O (2 m ) o maybe O (m 2 ), depending on how M {v} is stoed. Fo an inne node t labeled o + with the vetex set V t, the set of admitted matices M Vt can be calculated in time 2 O(m) if the sets of admitted matices of the child nodes ae known. This was poven by Fede et al. ([FHH06], Lemma 2.1 and its complement). Assume that the inne node t is a spide node labeled o, so the vetex set V t assigned to t is V t = {c 1,..., c ν, s 1,..., s ν } R. The set of admitted matices M Vt with lists L Vt can be calculated in time O (ν m 4 m + m 2 64 m ) if the set of admitted matices M R is known. This is the esult of Poposition 4. Because t has 2ν leaves as child nodes and the whole tee contains n leaves, the complexity fo calculating all sets of admitted matices fo all spide nodes is O (n m 4 m + m 2 64 m ) 2 O(m) n. The tee contains n leaves and theeby at most n 1 inne nodes, so calculating the sets of admitted matices fo evey node in the tee takes the time 2 O(m) n. With this infomation available, the list M-patition poblem can be decided since G with lists L admits M if and only if M M V and M V was calculated as the set of admitted matices fo the oot node. Coollay 3. The M-patition poblem with lists can be decided in linea time fo P 4 -spase gaphs if the size of M is constant The list M-patition poblem with vaiable-sized M If both the matix M and the gaph G with lists L ae vaiable, the list M-patition poblem is NP-complete, since 3-Satisfiability (3SAT) can be educed to the poblem. This fact will be shown in this section. The 3SAT poblem is about deciding whethe an abitay boolean fomula in conjunctive nomal fom (CNF) such that evey clause contains exactly thee liteals is satisfiable. This poblem is NP-complete [Coo71]. A boolean fomula is in CNF if each clause is a logical OR 20

25 3. List M-patitions of P 4 -spase Gaphs of its liteals and the clauses ae connected by a logical AND. A boolean fomula is satisfiable if the vaiables can be assigned values of tue and false such that the fomula evaluates tue. Let E be a boolean fomula in CNF with thee liteals pe clause. Let A N be the numbe of vaiables appeaing in E and let B N be the numbe of clauses. Let a 1,..., a A denote these vaiables. E = B ( 1 x β 2 x β 3 x β ) β=1 Evey i x β (β N B, i {1, 2, 3}) is a liteal, a possibly negated boolean vaiable of E. Vaiables can appea moe than once in the fomula, that means i x β1 and j x β2 might be the same vaiable, even if i j o β 1 β 2. Let G = (A B, {x y x, y A B x y}) be the complete gaph with A + B (A A, B B ) vetices. The vetex a α A (1 α A) epesents the vaiable a α of E. As explained late in detail, evey vetex in A will have two possible pats of an M-patition that detemine whethe the coesponding vaiable is tue o false in a specific assignment. Each vetex b β B (1 β B) coesponds to one clause 1 x β 2 x β 3 x β of E. These vetices choose between thee pats of an M-patition, whee evey pat equies a diffeent liteal of the clause to be tue. This will also be descibed moe fomally late. The symmetic matix M {0, 1, } 2 A+3 B 2 A+3 B has 2 A + 3 B ows and the same numbe of columns. M splits into two diagonal blocks 1 2A 2A M 2A and 1 3B 3B M 3B containing only 1 enties and a submatix C 2A M 3B as well as C s tansposed C T in the uppe ight and lowe left cone. Theefoe, it looks like this: ( 1 2A C C T 1 3B ) The columns of C can be gouped into B submatices β C {0, 1, } 2A 3 (1 β B). The matix βc encodes the clause 1 x β 2 x β 3 x β : Fo 1 α A, the i-th column (1 i 3) of the (2α 1)-th and (2α)-th ow of β C ae ( 1 1 ) if the liteal ix β does not epesent the vaiable a α, ( 1 0 ) if the liteal ix β unnegatedly epesents the vaiable a α, and ( 0 1 ) if the liteal ix β negates the vaiable a α. The vetex a α is only allowed to be placed in patitions 2α 1 and 2α by its list L (a α ) = {2α 1, 2α}. The vetex b β must be in only one of the patitions 2A + 3β 2, 2A + 3β 1, o 2A + 3β because of its list L (b β ) = {2A + 3β 2, 2A + 3β 1, 2A + 3β}. As the constuction of G, L and M is staightfowad, it is obviously possibly to constuct them in polynomial time, given a boolean fomula E. The following theoem educes the 21

26 3. List M-patitions of P 4 -spase Gaphs satisfiability poblem fo E to the list M-patition poblem and theefoe shows that it is NPcomplete. Poposition 5. The peviously defined gaph G with lists L admits an M-patition if and only if the boolean fomula E is satisfiable. Poof. Assume that G = (A B, E) admits an M-patition A 1,..., A 2A, B 1,..., B 3B with lists L. B i is the (2A + i)-th pat of the patition. By the definition of the lists L, we obviously have A = 2A i=1 A i and B = 3B i=1 B i. Fo evey α {1,..., A}, let the vaiable a α have the value tue if a α A 2α 1 and false if a α A 2α (by constuction of the list L (a α ) = {2α 1, 2α}, thee ae no futhe possibilities). Fo evey β {1,..., B}, the vetex b β is in a pat B 3β 3+i with i {1, 2, 3} by constuction of the list L (b β ) = {2A + 3β 2, 2A + 3β 1, 2A + 3β}. Let α {1,..., A} be chosen such that a α is the vaiable epesented by the liteal i x β, then is eithe ( 1 0 ) if ix β does not negate a α, o ( 0 1 ) if ix β negates a α. ( M 2α 1,2A+3β 3+i M 2α,2A+3β 3+i ) = ( C 2α 1,3β 3+i C 2α,3β 3+i ) = ( β C 2α 1,i βc 2α,i ) As the vetices b β and a α ae adjacent, a α must be in pat 2α 1 if the liteal i x β does not negate a α, in this case let a α be assigned tue, o in patition 2α if the liteal i x β negates a α, then let a α be assigned false. In both cases, i x β evaluates tue, so the whole clause 1 x β 2 x β 3 x β is tue with the given assignment. This applies to evey clause and theefoe, the assignment satisfies the boolean fomula E if the gaph G with lists L admits an M-patition. Assuming thee is an assignment to the vaiables a 1,..., a A satisfying E, an M-patition A 1,..., A 2A, B 1,..., B 3B admitted by G can be constucted: Fo evey α {1,..., A}, if a α is tue, then A 2α 1 {a α }, A 2α, if a α is false, then A 2α 1, A 2α {a α }. Fo evey β {1,..., B}, the clause 1 x β 2 x β 3 x β holds tue fo the obseved assignment. Theefoe, an i {1, 2, 3} exists such that i x β is tue. Afte defining B 3β 3+i {b β }, B 3β 3+ j ( j {1, 2, 3} {i}), all vetices ae assigned to pats with espect to thei lists. Since the diagonal blocks of M will obviously not pevent G fom admitting the M-patition (they contain only 1 enties and G is a complete gaph), only the submatix C needs to be 22

27 3. List M-patitions of P 4 -spase Gaphs analyzed moe thooughly. G would obstuct the M-patition with lists L only if thee wee α N A, β N B with a α A 2α i fo some i {0, 1} and b β B 3β 3+ j fo some j {1, 2, 3} such that C 2α i,3β 3+ j = 0. By the constuction of C, this can happen only if eithe i = 0 and the liteal jx β unnegatedly epesents the vaiable a α o i = 1 and the liteal j x β negates the vaiable a α. By the definition of the pats A 2α 1 and A 2α, a α would be false if j x β negates a α and tue if j x β is a α. In both cases, the liteal j x β evaluates false. This contadicts the choice of j, since, by the constuction of B 3β 3+ j, b β B 3β 3+ j implies that i x β is tue. Theefoe G with lists L admits the M-patition. Theoem 3. The list M-patition poblem is NP-complete fo complete gaphs if the gaph, its lists and M ae vaiable. Poof. The boolean fomula E is an abitay boolean fomula in CNF with exactly thee liteals pe clause. Deciding its satisfiability is the NP-complete 3SAT poblem [Coo71]. The peceding poposition theefoe educes the 3SAT poblem to a list M-patition poblem, so this is also NP-had. Whethe given sets ae a valid patition can be decided in polynomial time by compaing the adjacency of evey pai of vetices with the matix enty detemined by thei pats. Thus, the list M-patition poblem is in NP and so the list M-patition poblem is NP-complete fo complete gaphs. The theoem is still tue if G and M ae complemented: Coollay 4. The list M-patition poblem is NP-complete fo empty gaphs if the gaph, its lists and M ae vaiable. As the poblem is NP-complete fo complete gaphs, of couse it is also NP-complete fo all lage classes of gaphs. Coollay 5. If the gaph, its lists and M ae vaiable, the list M-patition poblem is NPcomplete fo complete bipatite gaphs, fo cogaphs, and fo P 4 -spase gaphs Maximum size of a minimal M-obstuction gaph with lists Theoem 4. Let f m 4 m+1 (m + 1)!. Fo evey m N, evey m-by-m matix M, and evey P 4 -spase gaph G with lists that is a minimal M-obstuction, G has at most f (m) vetices. Poof. Let M be a matix an m-by-m matix and let G be a P 4 -spase gaph with lists L that is a minimal M-obstuction. Eithe G o its complement is disconnected o G is a spide gaph since G is P 4 -spase. The theoem will be poved by induction ove m N. 23

28 3. List M-patitions of P 4 -spase Gaphs Let m = 1. If a gaph G obstucts M with lists L, then G = (V, E) has a vetex u V with the list L(u) = o G contains two vetices v, w V that ae adjacent o non-adjacent while the matix is M = (0) o M = (1), espectively. In the fist case, G {u} obstucts M with lists L {u} and since G is a minimal M-obstuction with lists L, we have V = {u} and G = 1 < f (1). In the second case, G {v, w} obstucts M with lists L {v,w} and this implies G = 2 < f (1). Now assume that the theoem is coect fo all m < m N {1}. Case 1: Assume that G o its complement is disconnected. If G is not a spide gaph, we may assume that G = G 1 G 2 is disconnected by complementing G and M if necessay. Fo j {1, 2}, let M j contain all patition submatices of M that G j = (V j, E j ) obstucts with lists L G j, all of which have a size lowe than m because G j = (V j, E j ) admits M with lists L G j. Using M {M} in Coollay 1 and Lemma 2 shows that G j is a minimal M j -obstuction with lists L G j. Fo evey matix M M j, the lagest P 4 -spase gaph that minimally obstucts M with lists has a size of at most f (i) by the inductive assumption whee i is the size of M. Fo each M M j, G j obstucts M with lists and theefoe contains a minimal M-obstuction G j, M = (V j, M, E j, M ) as an induced subgaph. G j ( M M j V j, M ) obstucts M j and as G j is a minimal M j -obstuction, we see V j = M M j V j, M. This implies G j M M j V j, M 0<i<m ( m ) i f (i). Combining the esults fom j = 1 and j = 2 yields G = G 1 + G 2 2 0<i<m ( m ) i f (i). By the inductive assumption, this implies the poposition: G 2 ( m 0<i<m i ) f (i) = 2 0<i<m m! i! (m i)! 4i+1 (i + 1)! = 2 (m + 1)!4 m+1 4 i m 0<i<m (m i)! < 4 m+1 4 k (m + 1)! 2 0<k<m k! < 4 m+1 (m + 1)! 2(e 1 4 1) < 4 m+1 (m + 1)! = f (m) i + 1 m + 1 Case 2: Assume that G is a spide gaph. G = (V, E) is a spide gaph with the vetex set V = S C R patitioned into the spide s body C = {c 1,..., c ν }, legs S = {s 1,..., s ν }, and head R. As G is a minimal M-obstuction with lists L, Lemma 3 implies that, fo each pai of lists (L(c i ), L(s i )) N m N m, thee ae at most m + 1 pais of vetices in {(c 1, s 1 );... ; (c ν, s ν )} that have these lists. Thee ae (2 m ) 2 possible pais of lists, so we see S C (m + 1) 2 (2 m ) 2. Fo R =, this implies the claim G = S C 2(m + 1)4 m < f (m). Thus, assume R. Fo each R, the induced subgaph G {} admits M with lists L G {} since G is a minimal M-obstuction with lists L. By Lemma 4, thee ae sets S, R N m such that G (R {}) with 24

29 3. List M-patitions of P 4 -spase Gaphs lists L R {} admits M R and G (S C) admits M S with lists L S C P(N m) defined as follows: L (c i ) L(c i ) {x N m y R.M x,y = 0} L (s i ) L(s i ) {x N m y R.M x,y = 1} As G obstucts M with lists L, by Lemma 4 it is not possible that G R admits M R with lists L R while G (S C) admits M S with lists L. Thus, G R obstucts M R with lists L R. This implies R N m since G R as an induced subgaph of the minimal M-obstuction G admits M = M N m. Let M R {M R R}. Then G R obstucts M R with lists L R and, fo any R, G (R {}) admits a matix in M R. Hence, G R is a minimal M R -obstuction. Simila to Case 1, we see that the size of G R is at most the sum of the sizes of all minimal M R -obstuctions. The matices in M R have a size of at most m 1, so we conclude R ( m 0<i<m i ) f (i) < 4 m+1 (m + 1)! (e 1 4 1) < 4 m+1 (m + 1)! 1 3 The size of G theefoe is G = S C + R = 2(m + 1)4 m + 4 m+1 (m + 1)! 1 3 = (m + 1)4 m m+1 (m + 1)! 1 3 < (4 m+1 (m + 1)!) ( ) < 4 m+1 (m + 1)! = f (m) 25

30 4. Minimal M-obstuction, P 4 -spase Gaph without lists This chapte has its focus on the M-patition poblem without lists. The matix M has no diagonal s, since othewise any gaph would admit M and the poblem becomes tivial. By pemutating the ows and columns, M may be witten in block fom with k diagonal 0s in the fist ows and then l diagonal 1s Uppe Bounds fo minimal M-obstuction, P 4 -spase Gaphs with bound clique sizes Poposition 6. Let M m µ=1 {0, 1, }µ µ be a non-empty set of matices with a maximum size of m N. Let G be a P 4 -spase gaph that obstucts M minimally. Let the maximum clique size in G be N. Then G has at most g(m, ) 2( m+ ) m 1 vetices. Poof. The poposition will be poved by induction ove m +. The lowest possible value fo m is 1, in which case thee is only one matix M M, and then G has at least one vetex, as it is an M-obstuction. This implies 1. The lowest possible value fo m + theefoe is m + = 2 with M = {(1)} and G = ({v, w}, }. This implies G = 2 = = g(1, 1). Let m, N and assume the poposition to be tue fo all, m with + m < + m. Let M be a set of matices with maximum size m and let G be a P 4 -spase gaph with maximum clique size. The oot node of G in its tee epesentation is eithe one of the two types of spide nodes o, a node fo a disjoint union, o a node + fo a join. The following poof distinguishes between these cases. The uppe limit of vetices of G fo oots, and + will be shown to be G g(m, t) + g(m, t) with t {1,..., 1}. If the oot is labeled, the uppe limit is only G g(m 1, ) + g(m, t) + g(m, t). 26

31 4. Minimal M-obstuction, P 4 -spase Gaph without lists Case 1: Assume the oot node is labeled +. In this case, thee ae two disjoint, P 4 -spase gaphs G 1, G 2 with G = G 1 + G 2. Let t be the size of the lagest clique in G 1 and let t be the lagest clique in G 2. The union of these cliques is a clique in G of size t + t. t + t being the size of the maximum clique in G implies 1 t 1 and t t. Using Lemma 2, we conclude that G 1 and G 2 ae minimal obstuction gaphs of matix sets with maximum size m, which means G g(m, t) + g(m, t) by induction. Case 2: Assume the oot node is a spide node. Befoe the main pat of the poof, note that the following fomula is coect fo all m 1 and ρ 1: 2ρ = 2ρ = 2( 1 + ρ ρ ) 1 1 2(m + ρ ) m 1 = g(m, ρ) (4.1.1) ρ Since G = (V, E) is a spide gaph, its node set may be witten as V = {c 1,..., c ν } {s 1,..., s ν } R, whee {c 1,..., c ν } is the spide s body, {s 1,..., s ν } is the spide s legs, and R is the spide s head, which is the same teminology as in Definition 11. If R is empty, G has exactly 2ν 2 vetices. Using Inequality 4.1.1, this esults in G 2 = 2 2t + 2t g(m, t) + g(m, t) fo any t {1,..., 1}. Now assume R. Any clique in G R is pat of a clique in G with the ν additional vetices c 1,..., c ν. As the maximum clique size in G is, the maximum clique size in G R is ν. By the inductive assumption, the numbe of vetices in G R is R g(m, ν), which limits the numbe of vetices in G to G = G {s 1,..., s ν, c 1,..., c ν } + G R = 2ν + R g(m, ν) + g(m, ν) The inequality in the peceding fomula is a esult of Inequality By the definition of spide nodes, we have 1 ν, and as the maximum clique size ν of G R is positive, we have ν <. Defining t ν esults in the same inequality as in the case R = : G g(m, t) + g(m, t) with 1 t 1. Case 3: Assume the oot node is labeled. Fo all 1 <, g(m, ) = 2( m+ ) m 1 < 2( m+ ) m 1 = g(m, ) is tue. Theefoe and by the inductive assumption, we conclude G < g(m, ) if the lagest clique in G contains less than vetices. Hence, we will assume that the lagest clique in G contains exactly vetices. G is a P 4 -spase gaph and theefoe pefect and -coloable. Let F {0, 1, } be the matix with only 0s on the diagonal and only s eveywhee else. Due to the fact that the gaphs admitting F ae exactly the -coloable gaphs and since G obstucts all matices in M, no matix in M contains F as a submatix. Since the oot node is labeled, G is a disconnected gaph. Let G 1 be the induced subgaph of G containing only the vetices of the connected component with the lagest clique, which has size, while the induced subgaph G 2 contains all othe vetices. Thus, G 1 and G 2 ae two disjoint, 27

32 4. Minimal M-obstuction, P 4 -spase Gaph without lists P 4 -spase induced subgaphs of G with G = G 1 G 2. As G obstucts the set M minimally, G 1 G admits a matix M M. Let M M be the set of patition submatices M Q of M such that thee ae sets of pats P, Q N m with M P,Q not containing a 1 and G 1 having an M P -patition. G 2 obstucts M M as othewise G 1 G 2 = G would admit M by Lemma 1. Because F is not a submatix of M and G 1 contains a clique of size, evey M P contains a 1 in its diagonal. Because M P,Q does not contain a 1, we conclude M Q M and theefoe M Q has a size smalle than m. G 2 obstucts M 2 M M M M, as it is a union of matix sets obstucted by G 2. Assume thee is an induced subgaph G 2 of G 2 that obstucts M 2. Let M be any matix in M. Then fo all sets of pats P, Q N m fo which G 1 admits M P and M P,Q contains no 1, we have M Q M 2 and theefoe G 2 obstucts M Q. This means G 2 G 1 obstucts M by Coollay 1. As M was chosen abitaily, G 2 G 1, an induced subgaph of G, also obstucts M. This contadicts that G is a minimal M-obstuction. Hence, G 2 is a minimal M 2 -obstuction and the matices in M 2 have a size of at most m 1. By the inductive assumption, G 2 has only G 2 g(m 1, ) vetices. Since G 1 is P 4 -spase and connected, the oot node in the tee epesentation of G 1 is eithe labeled + o a spide node. As shown, thee is a t {1,..., 1} in these cases such that G 1 has G 1 g(m, t) + g(m, t) vetices. Adding the two inequalities yields G = G 2 + G 1 g(m 1, ) + g(m, t) + g(m, t). The est of this poof equies an inequality that will be poved x+1 ) (a+x ), x+1 we find (a+x x ) All cases: Inductive step. fist. Iteatively using the well-known identity a, x N.( a+x x ) = (a+x+1 ( a+x x ) = x 1 i=x (a+i i+1 ) fo all x > x. Thus, fo t 1 and m N we have ( m + t m + ( t) ) + ( ) t t =( m ) (m ) + (m + t m + ( 1) m + ( 1) m + ( t) ) + ( ) ( ) + ( ) t 1 1 t =( m + 1 t 1 + ( 1) ) + (m ) + ( m + i i + 1 ) ( m + i i= t i + 1 ) =( m ( 1) ) + (m 1 ( m ( 1) ) + (m ) 1 1 i=1 t 1 ) + i=1 (( m + i i + 1 ) (m + i + t 1 )) i + t (4.1.2) Even in the wost case if the oot node is labeled, the poposition can be poved with the 28

33 4. Minimal M-obstuction, P 4 -spase Gaph without lists help of Inequality 4.1.2: G g(m 1, ) + g(m, t) + g(m, t) = 2( m + 1 ) m 2 + 2( m + t ) m 1 + 2( m + t ) m 1 t t 2 (( m + 1 ) + ( m )) + 2(m 1 1 ) 3m 4 = 2( m + ) + (2m + 2) 3m 4 < 2( m + ) m 1 = g(m, ) Complementing G in Poposition 6 yields the following coollay: Coollay 6. Let M be a matix set with maximum matix size m N. Let G be a P 4 -spase gaph and a minimal M-obstuction. The lagest stable set in G contains at most vetices. Then G has at most g(m, ) = 2( m+ ) m 1 vetices Uppe Bounds fo minimal M-obstuction, P 4 -spase Gaphs Theoem 5. Any minimal M-obstuction, P 4 -spase gaph G has at most O (16 m ) vetices. Poof. Let G be in its tee epesentation and assign evey node t of the tee a set M t of patition submatices of M such that the gaph G t coesponding to the node t is a minimal M t - obstuction. The set M t0 of the oot t 0 consists of the matix M only. Obviously, the numbe of vetices of G t is the numbe of leaves of the subtee ooted at t and theefoe, the size of G = G t0 is the numbe of leaves in the whole tee. Let h (m, m) denote the maximum numbe of vetices that a gaph G t can have if the geatest size of a matix in M t is m and all matices in M t ae submatices of a matix of size m. In the fist step of this poof, the inequality h (m, m) 2 m 2 (2( ) m m m 1) + 2h (m, m 1) will be shown. 2 If G t has a maximum clique size of at most m, then the size of G t is G t 2( ) m m m 1 by Poposition 6. The size of G t is also G t 2( 2 m m ) m 1 if the lagest independent set in G t has size at most m by Coollay 6. If instead the maximum clique size of G t is geate than the maximum matix size m of M t and the maximum independent set size of G t is also geate than m, thee ae diffeent cases to conside: 29

34 4. Minimal M-obstuction, P 4 -spase Gaph without lists Case 1: Assume the node t is labeled. Let k be the maximum numbe of diagonal 0s of the matices in M t. The node t has two child nodes t and t with G t = G t G t. Thee ae two subcases, of which only the fist will be poved completely hee. In the second subcase, G t will be shown to consist of the induced subgaphs G with an independent set of size geate than m and an induced subgaph G with G 2 ( ) m m m 1. The second subcase will be shown late to also lead to a numbe of vetices of G t of at most 2 m 2 (2( ) m m m 1) + 2h (m, m 1). In the fist subcase, G t and G t both contain a clique of size geate than k. Let M t be the set of all matices M Q such that thee is a matix M M t with size ˆm and and pats P, Q N ˆm such that M P,Q contains no 1 and G t admits M P. Thee ae moe vetices in the lagest clique of G t than thee ae diagonal 0s in M P. Because vetices of the same clique cannot be placed in the same pat i if M i,i = 0, evey M P -patition of G t uses a pat j P with M j, j = 1. Fom the facts that M P,Q contains no 1, M j, j = 1 and j P follows j Q, and theefoe the matix M Q is smalle than M P Q and especially Q < ˆm m. By Coollay 1, G t obstucts the matix M Q and theefoe G t obstucts M t. By Lemma 2, G t is also a minimal M t -obstuction. As the maximum matix size of M t is smalle than m, this means G t h (m, m 1). Because G t also contains a clique of size geate than k, the gaphs G t and G t in the poof above can be exchanged, which poves the inequality G t h (m, m 1). Theefoe, the size of G t is G t = G t + G t 2h (m, m 1). In the othe subcase, let G G t have a maximum clique size of at most k, without loss of geneality, and let the lagest clique in G G t have a size geate than m. Theefoe the numbe of vetices in G t is G 2 ( ) m m m 1. Obviously no matix can contain moe diagonal 0s than it has ows, so k is smalle than m. This subcase will be analyzed togethe with simila subcases of diffeent labels fo the node t late in this poof. Case 2: Assume the node t is labeled +. Let l be the maximum numbe of diagonal 1s of the matices in M t. Let t and t be the child nodes of t, which means that G t = G t + G t. Complementing G and M in the case whee t is labeled leads to the conclusion that G t has at most 2h (m, m 1) vetices if both G t and G t contain an independent set of size geate than l. Othewise and without loss of geneality, let the gaph G G t have a maximum independent set with size of at most l and theefoe its numbe of vetices is G 2 ( ) m m m 1. In this subcase, G G t has an independent set of size geate than m. This subcase will be analyzed late in the poof, togethe with simila subcases of diffeent labels fo the node t. Case 3: Assume the node t is a spide node. Using the teminology fom Definition 11, let G t = (V t, E t ) contain the vetices in the set V t = {c 1,..., c ν } {s 1,..., s ν } R. Because G t is a minimal M t -obstuction gaph, Lemma 3 can be applied if lists ae defined as L (v) V t {v} fo evey vetex v V t, which effectively means that thee ae no estictions whee vetices can be placed. Lemma 3 implies that the numbe 2ν of vetices in the spide s body and legs G t R is at most 2 m

35 4. Minimal M-obstuction, P 4 -spase Gaph without lists If the spide s head G t R has a maximum clique size o maximum independent set size of at 2 most m, then the numbe of vetices of G t R is at most 2( ) m m m 1. In this subcase, the 2 numbe of vetices in G t is G t = 2ν + R 2 m ( ) m m m 1 = 2(2 ) m m + m + 1. Othewise, G G R has a clique and an independent set of size geate than m and G G R has at most 2 m+2 vetices. This subcase will now be analyzed togethe with the simila subcases of nodes labeled o +. 2 At this point, G t has been shown to eithe have less than max {2( ) m m + m + 1, 2h (m, m 1)} vetices o that all of the vetices of G t ae in the disjoint induced subgaphs G and G, whee G has at most max {2 m + 2, ( 2 m m ) m 1} vetices and G has an independent set o clique of size geate than m. G 1 G may have less than max {2( 2 m m ) + m + 1, 2h (m, m 1)} vetices o, again, G 1 splits up into two induced subgaphs G 2 and G 2 with G 2 being a gaph with an independent set o clique of size geate than m and G 2 having at most max{2 m+2, (2 ) m m m 1} vetices. Futhe epetition leads to 2s gaphs G 1,...,G s and G 1,...,G s, such that fo evey 1 i s, G i contains a clique and an independent set with moe than m vetices and G 2 i has at most max{2 m + 2, ( ) m m m 1} vetices. Let G 2 s have at most max {2( ) m m + m + 1, 2h (m, m 1)} vetices. An uppe limit fo s will now be poved, namely s 2 m, which will eventually lead to an uppe limit fo the numbe of vetices of G t. Fo 1 i s, let N i be the set N i {P N m G i obstucts M P}. Fo two indices i, j (1 i < j s), G j is an induced subgaph of G i and G i is a minimal obstuction gaph fo some set M t of submatices of M. Theefoe G j does not obstuct M t and hence N i N j. As thee ae only 2 m diffeent subsets of N m, this implies s 2 m, Hence, the s gaphs G 1,...,G s togethe have at most 2 m 2 max{2 m + 2, ( ) m m m 1} vetices while G s has at most max {2( 2 m m ) + m + 1, 2h (m, m 1)} vetices. Because evey vetex of G t lies in exactly one of the induced subgaphs G 1,...,G s,g s, G t has at most 2 m max{2 m + 2 2, ( ) m m m 1} + max {2(2 ) m m + m + 1, 2h (m, m 1)} vetices, which togethe with the othe cases completes the fist step of the poof: Fo evey m m > 1, the maximum numbe of vetices of a minimal M-obstuction, P 4 -spase gaph whee all matices in M ae submatices of a matix M {0, 1, } m m is h (m, m) 2 m 2 max{2 m + 2, ( ) m 1} + max {2(2 ) + m + 1, 2h (m, m 1)} m m m m < (2 m 2 + 1) (2( ) + m + 1) + 2h (m, m 1) m m In the case m = 1, we have h (m, 1) 3 < (2 m + 1) (2( 2 1 ) ). In the second step of the poof, the ecusion will be esolved to a sum. The fomula will be simplified even futhe with the help of the inequality m m=1 (2 ) m m 2m m=1 (2m m ) = 22m : 31

36 4. Minimal M-obstuction, P 4 -spase Gaph without lists m h (m, m) < 2 m m (2 m 2 + 1) (2( ) + m + 1) m=1 m m (2 2m+1 m 2 ) (2( ) + m + 1) m=1 m m (2 2m+1 ) (2 2m+1 m(m + 1) + + m) 2 Theefoe, the numbe of vetices of any minimal M-obstuction, P 4 -spase gaph is at most h (m, m) O ((2 2m+1 ) (2 2m+1 + m(m+1) 2 + m)) = O(16 m ). 32

37 5. Minimal M-obstuction, P 4 -spase Gaphs with constant matices M In this chapte, let the matix M be an (a, b, c)-block matix. This means that its fist k diagonal enties ae all 0 and all othe l diagonal enties ae 1. The diagonal of M does not contain any, so the matix size is m = k + l. The enties of M can be divided into fou blocks: the uppe left block with all diagonal 0s, the lowe ight block with all diagonal 1s, and the two emaining blocks on the lowe left and the uppe ight. In each of these blocks, all enties have the same value, except those in the diagonal of M. The off-diagonal enties in the uppe left block all have the value a, while the off-diagonal enties in the lowe ight block all have the value b. All othe enties, which ae those in the uppe ight and lowe left block, have the value c. a, b, and c can be 0, 1, and, but we may assume a 0 and b 1 because such blocks ae equal to a single 0 and 1 enty. A matix meeting those citeia is also called a constant matix. An (a, b, c)-block matix is completely detemined by the paametes a, b, c, the numbe k of 0s in its diagonal and the numbe l of 1s in its diagonal. If a,b,c ae clea fom the context, then M[k, l ] stands fo the (a, b, c)-block matix with k diagonal 0s and l diagonal 1s. Note that M[k, l ] is a patition submatix of M if and only if k k and l l. If k < 0 o l < 0, then M[k, l ] shall be teated like a matix that evey gaph obstucts. Lemma 5. Let M be an (a, b, c)-block matix with a =, b {0, }, c {0, }, let G = G 1 G 2 be a disconnected gaph and let k, x, y N be natual numbes. If G 1 admits M[k, x] and G 2 admits M[k, y] then G admits M[k, x + y]. Poof. Let A 1,..., A k+x be an M[k, x]-patition of G 1 and let B 1,..., B k+y be an M[k, y]-patition of G 2. Then obviously A 1 B 1,..., A k B k, A k+1,..., A k+x, B k+1,..., B k+y is an M[k, x + y]- patition of G (,, )-block matices Theoem 6. Let M be a (,, )-block matix and let G be a minimal M-obstuction, P 4 -spase gaph. Then G is a cogaph. Poof. This poof distinguishes between two cases: Eithe one of the values k o l ae zeo o both k and l ae geate than 0. 33

38 5. Minimal M-obstuction, P 4 -spase Gaphs with constant matices M Stating with the fist case, eithe k o l is 0. We may assume l = 0, the othe subcase follows by complementation of G and M. A gaph admits M if and only if it is k-coloable. Since G as a P 4 -spase gaph is pefect, it is k-coloable if and only if its lagest clique consists of at most k vetices. Theefoe, a P 4 -spase gaph obstucts M if and only if it has the complete gaph K m+1 with m + 1 = k + 1 vetices as an induced subgaph. K m+1 is P 4 -spase aleady, so it is the only minimal M-obstucting P 4 -spase gaph. K m+1 is also a cogaph, so all minimal M-obstuction P 4 -spase gaphs ae cogaphs if k = 0 o l = 0. Looking at the second case, let both k and l be at least 1. Fist assume the oot node in the tee epesentation of G is a spide node. Using the teminology as in Definition 11, the node set V of G = (V, E) may be witten as V = {c 1,..., c ν, s 1,..., s ν } R whee {c 1,..., c ν } is the spide s body, {s 1,..., s ν } is the spide s legs, and R is the spide s head. Since G is a minimal M-obstuction, evey induced subgaph of G admits M, so thee is an M-patition A 1,..., A m of G R, whee A 1,..., A k ae independent sets and A k 1,..., A m ae l cliques. The vetices s 1,..., s ν ae non-adjacent to each othe and they ae non-adjacent to evey vetex in R, so A 1 {s 1,..., s ν } is also an independent set. Similaly, the vetices c 1,..., c ν ae adjacent to each othe and adjacent to evey vetex in R, so A m {c 1,..., c ν } constitutes a clique. Hence, A 1 {s 1,..., s ν }, A 2,..., A m 1, A m {c 1,..., c ν } is an M-patition of the M-obstucting gaph G, which is a contadiction. Theefoe the oot node of G is not a spide node. The est of the poof uses induction on the matix size m. The induction base m = 1 has aleady been poved, as this case implies eithe k = 0 o l = 0. The induction step was aleady poved fo all matices smalle than the given matix M. As shown above, the oot node in the tee epesentation of G is not a spide node, so by complementing G and M if necessay, we may assume that G = G 1 G 2 is a disconnected gaph. G is a minimal M-obstuction, so the induced subgaphs G 1 and G 2 both admit M = M[k, l]. This implies that G 1 obstucts M[k, 0], as othewise G would admit M[k, 0 + l] = M[k, l] = M by Lemma 5. Now let l 1 be the smallest numbe such that G 1 admits M[k, l 1 ]. As G 1 obstucts M[k, 0] but admits M[k, l], we have 0 < l 1 l. G obstucts M = M[k, l 1 + (l l 1 )] and hence G 2 obstucts M[k, l l 1 ]. Let M 1, M 2 be matix sets defined as M 1 {M[k, l ] 0 l < l 1 } and M 2 {M[k, l ] 0 l l l 1 }. This satisfies the conditions of Lemma 2: G 1 obstucts M 1 and G 2 obstucts M 2. Fo any two sets of pats P, Q N m with M P M 1 and M Q M 2, the matix M P has the patition submatix M[0, l 1 ] and M Q has the patition submatix M[0, l l 1 +1]. Since M P and M Q togethe contain at least l 1 + l l > l ows with a 1 in the diagonal enty, the submatix M PQ of M contains at least one 1 in the diagonal. By Lemma 2, G 1 is a minimal M 1 -obstuction and G 2 is a minimal M 2 -obstuction. All matices obstucting M[k, l 1 1] also obstuct M 1 and all matices obstucting M[k, l l 1 ] also obstuct M 2. Thus, G 1 is a minimal M[k, l 1 1]-obstuction and G 2 is a minimal M[k, l l 1 ]-obstuction, so by the inductive assumption, both G 1 and G 2 ae cogaphs. This makes G = G 1 G 2 a cogaph, too. Because Fede et al. have detemined the numbe of vetices of cogaph minimal M-obstuctions fo (,, )-block matices ([FHKM99], Coollay 4.2), the numbe of vetices of G can be exactly specified using the peceding theoem: 34

39 5. Minimal M-obstuction, P 4 -spase Gaphs with constant matices M Coollay 7. Let M be a (,, )-block matix. Then evey P 4 -spase, minimal M-obstuction gaph has exactly (k + 1)(l + 1) vetices Staicase-like (a, b, c)-block matices By excluding the edundant cases a 0 and b 1, thee ae 12 combinations of a, b, and c to conside. In the fist of these combinations, (a, b, c) = (,, ), minimal M-obstucting P 4 - spase gaphs wee aleady shown to have the same uppe bound of vetices as M-obstuction cogaphs in Theoem 6. In this section, uppe bounds fo the othe combinations will be poved. In ode to do that, spide gaphs and disconnected gaphs will be shown to have a couple of popeties that ae needed fo the poof of the uppe bound. Lemma 6. Let M be an (1,, )-block matix and let G be an M-obstuction spide gaph with the vetex set V = {c 1,..., c ν, s 1,..., s ν } R whee {c 1,..., c ν } is the spide s body, {s 1,..., s ν } is the spide s legs, and R is the spide s head. Then all of the following holds: G R obstucts M[k, l ν] k = 0 o l = 0 o G R obstucts M[1, l]. Poof. We assume that G R does not obstuct M[k, l ν], so thee is an M[k, l ν]-patition A 1,..., A k, B 1,..., B l ν of G R. The vetices c 1,..., c ν, s 1,..., s ν can be patitioned into the ν cliques C i {c i, s i } (1 i ν) if G is a slim spide o C 1 {c 1, s ν }, C i {c i, s i 1 } (2 i ν) if G is a fat spide. Then A 1,..., A k, B 1,..., B l ν,c 1,...,C ν ae an M[k, l ν+ν]-patition, which contadicts the fact that G obstucts M = M[k, l]. Figue 5.1 illustates this M[k, l]-patition of G in the case of a slim spide gaph. Assume next that k > 0 and l > 0 although G R does not obstuct M[1, l]. Then thee is an M[1, l]-patition A, B 1,..., B l of G R such that A is an independent set and B 1,..., B l ae cliques. Since the vetices s 1,..., s ν ae non-adjacent to all vetices in R, especially A, and since the vetices c 1,..., c ν ae adjacent to all vetices in R, the set A {s 1,..., s ν } is an independent set in G and the set B 1 {c 1,..., c ν } is a clique in G. Theefoe, A {s 1,..., s ν }, B 1 {c 1,..., c ν }, B 2,..., B l is an M[1, l]-patition of G. Since k > 0, the matix M[1, l] is a patition submatix of M[k, l] and theefoe G does not only admit M[1, l] but also M[k, l] = M, which is a contadiction. Figue 5.2 illustates this M[1, l]-patition of G in the case of a fat spide gaph. Poposition 7. Let M be an (1,, )-block matix with l > 0 and let G be a spide gaph with the vetex set V = {c 1,..., c ν, s 1,..., s ν } R whee {c 1,..., c ν } is the spide s body, {s 1,..., s ν } is the spide s legs, and R is the spide s head. G obstucts M if and only if G {c 1,..., c ν } obstucts M. 35

40 5. Minimal M-obstuction, P 4 -spase Gaphs with constant matices M Figue 5.1.: Possible patition of G if G R does not obstuct M[k, l ν] Figue 5.2.: Possible patition of G if G R does not obstuct M[1, l] (IS is an independent set) Poof. Assume that G obstucts M. By Lemma 6, G R obstucts M[k, l ν] and, if k is geate than 0, G R also obstucts M[1, l]. We assume contay to the poposition that G G {c 1,..., c ν } admits M. Then thee is an M-patition A 1,..., A k, B 1,..., B l of G. Assume all vetices s 1,..., s ν in the spide s legs ae elements of clique pats. All vetices in the spide s legs ae paiwise non-adjacent, theefoe we may assume s 1 B 1,..., s ν B ν, without loss of geneality. As the vetices in R ae non-adjacent to the vetices in the spide s legs, none of the vetices in R can be an element of a clique togethe with a vetex s j (1 j ν). This implies B j R = (1 j ν) and theefoe the pats A 1 R,..., A k R, B ν+1 R,..., B l R constitute an M[k, l ν]-patition of G R. This contadicts Lemma 6. Assume that at least one of the vetices s 1,..., s ν in the spide s legs is an element of an independent set pat. This implies k > 0 and then, by Lemma 6, G R obstucts M[1, l]. Assuming s 1 A 1 without loss of geneality, as M is a (1,, )-block matix, all vetices in A 2,..., A k must be adjacent to s 1. Because s 1 is non-adjacent to the vetices in R, we get A j R = (2 j k). This makes A 1 R, B 1 R,..., B l R an M[1, l]-patition of G R, a contadiction to the initial assumption that G R obstucts M[1, l]. The initial assumption must be wong and G obstucts M. Tivial. Complementing M yields the following coollay: Coollay 8. Let M be a (, 0, )-block matix with k > 0 and let G be a spide gaph with the vetex set V = {c 1,..., c ν, s 1,..., s ν } R whee {c 1,..., c ν } is the spide s body, {s 1,..., s ν } is the spide s legs, and R is the spide s head. G obstucts M if and only if G {s 1,..., s ν } obstucts M. Lemma 7. Let M be an (a, b, c)-block matix with a = 1 and let the gaph G have an induced subgaph P 3 ({x, y, z}, {x z}). Then G obstucts M[k, 0]. 36

41 5. Minimal M-obstuction, P 4 -spase Gaphs with constant matices M Poof. Assume G had an M[k, 0]-patition A 1,..., A k. Because a = 1 and because thee ae no clique pats, two vetices ae adjacent to each othe if and only if they ae in diffeent independent set pats. The non-adjacent vetices x and y must be elements of the same independent set pat, let this pat be x, y A 1, without loss of geneality. z is adjacent to x, which implies z A 1, while it is non-adjacent to y, which implies z A 1. This is a contadiction and so G obstucts M[k, 0]. Coollay 9. Let M be an (a, b, c)-block matix with a = 1 and let G be a spide gaph with the vetex set V = {c 1,..., c ν, s 1,..., s ν } R whee {c 1,..., c ν } is the spide s body, {s 1,..., s ν } is the spide s legs, and R is the spide s head. Then G {c 1 } obstucts M[k, 0] and fo G P 4, the following statements ae tue: if G is a fat spide gaph, the induced subgaph G {s 1, c 1 } obstucts M[k, 0]. if G is a slim spide gaph, the induced subgaph G {s 1, c 2 } obstucts M[k, 0]. Poof. By Lemma 7, we have to show that these gaphs contain P 3 as an induced subgaph. G {c 1 } contains the induced subgaph G {c 2, s 1, s 2 } = P 3. If G is a fat spide gaph, G G {s 1, c 1 } contains the non-adjacent vetices s s 2 and c c 2. If G is a slim spide gaph, let the vetices s and c in G G {s 1, c 2 } be defined as s s 2 and c c 1, so they ae non-adjacent again. If thee is a vetex R, G contains the induced subgaph G {, s, c} = P 3, as shown in Figue 5.3. Fo R =, we must have ν > 2 as othewise G would be P 4. In this case, G contains G {s 2, s 3, c 3 } = P 3, which is shown in Figue 5.4. Figue 5.3.: Example of the induced subgaph G {s 1, c 1 } of a fat spide gaph G Figue 5.4.: Example of a slim spide gaph G P 4 and its induced subgaph G {s 1, c 2 } Combining the esults of Poposition 7, Coollay 8, and Coollay 9 as well as its complement yields the following poposition since evey M-obstuction spide gaph G has an M-obstuction induced subgaph G G: Poposition 8. Let M be a (a, b, c)-block matix with (a, b, c) {(1,, ), (, 0, )} and let G be a spide gaph. Then G is not a minimal M-obstuction. 37