The computational complexity of the parallel knock-out problem
|
|
- Polly Daniel
- 5 years ago
- Views:
Transcription
1 Theoretcal Computer Scence 393 (2008) The computatonal complexty of the parallel knock-out problem Hajo Broersma, Matthew Johnson, Danël Paulusma, Ian A. Stewart Department of Computer Scence, Durham Unversty, South Road, Durham, DH1 3LE, UK Receved 24 July 2007; accepted 25 November 2007 Communcated by D. Peleg Abstract We consder computatonal complexty questons related to parallel knock-out schemes for graphs. In such schemes, n each round, each remanng vertex of a gven graph elmnates exactly one of ts neghbours. We show that the problem of whether, for a gven bpartte graph, such a scheme can be found that elmnates every vertex s NP-complete. Moreover, we show that, for all fxed postve ntegers k 2, the problem of whether a gven bpartte graph admts a scheme n whch all vertces are elmnated n at most (exactly) k rounds s NP-complete. For graphs wth bounded tree-wdth, however, both of these problems are shown to be solvable n polynomal tme. We also show that r-regular graphs wth r 1, factor-crtcal graphs and 1-tough graphs admt a scheme n whch all vertces are elmnated n one round. c 2007 Elsever B.V. All rghts reserved. Keywords: Parallel knock-out; Graphs; Computatonal complexty 1. Introducton In ths paper, we consder parallel knock-out schemes for fnte undrected smple graphs. These were ntroduced by Lampert and Slater [9]. Such a scheme proceeds n rounds: n the frst round each vertex n the graph selects exactly one of ts neghbours, and then all the selected vertces are elmnated smultaneously. In subsequent rounds ths procedure s repeated n the subgraph nduced by those vertces not yet elmnated. The scheme contnues untl there are no vertces left, or untl an solated vertex s obtaned (snce an solated vertex wll never be elmnated). A graph s called KO-reducble or smply reducble f there exsts a parallel knock-out scheme that elmnates the whole graph. The parallel knock-out number of a graph G, denoted by pko(g), s the mnmum number of rounds n a parallel knock-out scheme that elmnates every vertex of G. If G s not reducble, then pko(g) =. Our man motvaton for studyng the concept of reducblty s ts ntmate relaton to well-studed concepts n structural and algorthmc graph theory, lke matchngs and cycles. To llustrate ths, we note that a graph G wth a A prelmnary verson of ths paper was presented at the 7th Latn Amercan Theoretcal Informatcs Symposum 2006 and an extended abstract appeared n Lecture Notes n Computer Scence 3887 (2006) pp Correspondng author. Tel.: E-mal addresses: hajo.broersma@durham.ac.uk (H. Broersma), matthew.johnson2@durham.ac.uk (M. Johnson), danel.paulusma@durham.ac.uk (D. Paulusma),.a.stewart@durham.ac.uk (I.A. Stewart) /$ - see front matter c 2007 Elsever B.V. All rghts reserved. do: /j.tcs
2 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) perfect matchng has pko(g) = 1, as each vertex can select the vertex t s matched wth n a perfect matchng of G. Smlarly, a graph G wth a hamltonan cycle has pko(g) = 1, as each vertex can select ts successor on a hamltonan cycle of G wth some fxed orentaton. Whereas t s easy to check (.e., by a polynomal algorthm) whether a graph admts a perfect matchng, t s NP-complete to decde whether a graph has a hamltonan cycle. What can be sad about the complexty of decdng whether a graph G has a fnte parallel knock-out number? Or about determnng (an upper bound on) the value of pko(g)? These complexty questons are our man concern n ths paper and wll be answered n Secton 4. We wll also consder several structural propertes related to reducblty, but only wth some relaton to complexty questons. Other structural propertes related to reducblty can be found n [3] and [5] Complexty questons related to reducblty Consder the followng decson problem. PARALLEL KNOCK-OUT (PKO) Instance: A graph G. Queston: Is G reducble? In [9], whch appeared n 1998, t was clamed that PKO s NP-complete even when restrcted to the class of bpartte graphs. No proof was gven; the reader was referred to a paper that was n preparaton. Our attempts to obtan and verfy ths proof have been unsuccessful. We shall obtan the result as a corollary to a stronger theorem (Theorem 1 below) by consderng a related problem, whch s defned for each postve nteger k. PARALLEL KNOCK-OUT (k) (PKO(k)) Instance: A graph G. Queston: Is pko(g) k? Our frst result classfes the complexty of PKO(k), k 2. Theorem 1. For k 2, PKO(k) s NP-complete even f nstances are restrcted to the class of bpartte graphs. The proof s postponed to Secton 4. By usng almost the same arguments, we wll also show that decdng whether pko(g) = k s polynomally solvable for k = 1 and NP-complete for any fxed k 2 that s not part of the nput. As a matter of fact t s not dffcult to show that a graph G has pko(g) = 1 f and only f G contans a [1, 2]-factor,.e., a spannng subgraph n whch every component s ether a cycle or an edge: smply note that a vertex u that selects a vertex v s ether selected by v or by a vertex w {u, v}, and combne ths observaton wth the fact that every vertex selects exactly one other vertex and that all the graphs we consder are fnte. The problem of decdng whether G contans a [1, 2]-factor s a folklore problem appearng n many standard books on combnatoral optmzaton. For convenence we shortly dscuss t below. Let V (G) = {v 1, v 2,..., v n }. Defne a bpartte graph G wth vertex set V (G ) = {u 1, u 2,..., u n, w 1, w 2,..., w n } n whch u w j E(G ) and u j w E(G ) f and only f v v j E(G). A [1, 2]-factor n G corresponds to a perfect matchng n G. Hence the related decson problem and consequently PKO(1), can be decded n polynomal tme. Ths s also clear from the followng polynomally solvable decson problem (see [7], problem GT13, page 193): gven a drected graph D, decde whether V (D) can be parttoned nto dsjont sets of cardnalty at least 2 such that each of the sets nduces a subgraph wth a drected hamltonan cycle. To show the ntmate relaton wth knock-out schemes, replace each edge of G by two oppostely drected arcs. Clearly G has a [1, 2]-factor f and only f the related drected graph has such a partton nto hamltonan cycles. In [3], t was shown, usng a dynamc programmng approach, that the parallel knock-out number for trees can be computed n polynomal tme. The authors presented an O(n 3.5 log 2 n) algorthm for computng the parallel knockout number of an n-vertex tree, and asked whether there exsts a substantally faster algorthm for ths problem, wth a tme complexty of, say, O(n log n) or O(n 2 )? Our next result mples that there exsts a lnear tme algorthm for ths problem. A key ngredent of the dynamc program for trees n [3] s the reducton to a number of polynomally solvable bpartte matchng problems. For hgher tree-wdths, these bpartte matchng problems have no natural polynomally solvable analogues. Therefore the dynamc program for trees does not carry over to the bounded tree-wdth classes.
3 184 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) In [3] t was asked whether one can avod the computaton of perfect matchngs n auxlary bpartte graphs whle computng pko(t ) for a tree T. And can one then generalze such a method to graphs of bounded tree-wdth? In our second result, we gve an affrmatve answer, although we do not provde an explct algorthm. Theorem 2. The problem PKO(k) can be solved n lnear tme on graphs wth bounded tree-wdth. We wll also show that PKO can be solved n polynomal tme on graphs wth bounded tree-wdth Structural propertes related to reducblty As noted above, there s an ntmate relatonshp between reducblty and other structural propertes, lke the exstence of a [1, 2]-factor, a noton that s a common generalzaton of a perfect matchng and a hamltonan cycle. Apart from hamltonan graphs and graphs that have a perfect matchng, for example also all k-traversable graphs have been shown to have a [1, 2]-factor [4]. A graph s k-traversable f t admts a closed walk n whch every vertex occurs exactly k tmes. These graphs were also studed n [8]. We establsh several other results n Secton 6 that explore the relatonshp between reducble graphs and propertes related to the exstence of near perfect matchngs or hamltonan cycles. For example we show that all factor-crtcal and all 1-tough graphs have a [1, 2]-factor,.e., have parallel knock-out number 1. We refer to Secton 6 for defntons. We also show there that r-regular graphs wth r 1 have a [1, 2]-factor, and we have a closer look at almost regular bpartte graphs,.e., bpartte graphs n whch all vertces n the same bpartton class have the same degree Organzaton of the paper In Sectons 2 and 3 we ntroduce a number of defntons and prelmnary observatons. In Sectons 4 and 5 are the proofs and corollares of Theorems 1 and 2, respectvely. Secton 6 deals wth factor-crtcal graphs, 1-tough graphs, r-regular graphs and almost regular bpartte graphs. 2. Prelmnares Graphs n ths paper are denoted by G = (V, E). An edge jonng vertces u and v s denoted uv. For graph termnology not defned below, we refer to [2]. For convenence we allow graphs to have an empty vertex set. We say that G = (V, E) s the null graph f V = E =. For a vertex u V we denote ts neghbourhood, that s, the set of adjacent vertces, by N(u) = {v uv E}. The degree d G (v) of a vertex v n G s the number of edges ncdent wth t, or, equvalently, the sze of ts neghbourhood. A maxmal connected subgraph of a graph G s called a component of G. Adoptng the termnology and notaton from [3], for a graph G, a KO-selecton s a functon f : V V wth f (v) N(v) for all v V. If f (v) = u, we say that vertex v fres at vertex u, or that vertex u s knocked out by vertex v. For a KO-selecton f, we defne the correspondng KO-successor of G as the subgraph of G that s nduced by the vertces n V \ f (V ); f H s the KO-successor of G we wrte G H. Note that every graph wthout solated vertces except for the null graph has at least one KO-successor. A graph G s called KO-reducble, f there exsts a fnte sequence G G 1 G 2 G r, where G r s the null graph. If no such sequence exsts, then pko(g) =. Otherwse, the parallel knock-out number pko(g) of G s the smallest number r for whch such a sequence exsts. A sequence of KO-selectons that transform G nto the null graph s called a KO-reducton scheme. A sngle step n ths sequence s called a round of the KOreducton scheme. A subset of V s knocked out n a certan round f every vertex n the subset s knocked out n that round. We make some smple observatons that we wll use later on. Observaton 3. Let G = (V, E) be a KO-reducble graph, and let V 1 = {v V d(v) = 1}. Then n the frst round of any KO-reducton scheme each vertex of V 1 s knocked out by ts unque neghbour n G.
4 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) Proof. Ths s clear, snce otherwse some vertex v V 1 wll be an solated vertex after the frst round, as the neghbour of v s knocked out by v n the frst round. Observaton 4. Let G be a graph on at least three vertces. If G contans two vertces of degree 1 that share the same neghbour, then G s not KO-reducble. Proof. Suppose G s KO-reducble. Then by Observaton 3, the shared neghbour knocks out both vertces of degree 1, a contradcton. Observaton 5. Let u 1, u 2, u 3, u 4 be four vertces of a KO-reducble graph G such that N(u 2 ) = {u 1, u 3 }, N(u 3 ) = {u 2, u 4 } and N(u 4 ) = {u 3 }. If u 1 s knocked out n the frst round of a KO-reducton scheme, then u 1 fres at u 2 n the frst round. Proof. By Observaton 3, u 3 and u 4 knock each other out n the frst round, so u 3 does not knock out u 2. If u 1 s knocked out n the frst round of a KO-reducton scheme, then u 1 fres at u 2 n the frst round; otherwse u 2 wll be an solated vertex after the frst round. An odd path u 1 u 2... u 2k+1 s called a centred path of G wth centrevertex u k+1 f G {u k+1 } contans as components the path u 1 u 2... u k and the path u k+2 u k+3... u 2k+1. Observaton 6. Let P = u 1 u 2... u 7 be a centred path of a KO-reducble graph G. In the frst round of any KOreducton scheme u 1 and u 2 fre at each other, u 3 fres at u 2, u 6 and u 7 fre at each other, u 5 fres at u 6, u 4 fres at u 3 or u 5, and u 4 wll not be knocked out. In the second round of any KO-reducton scheme u 4 and ts remanng neghbour n P fre at each other. Proof. By Observaton 3, u 1 and u 2, and u 6 and u 7 knock each other out n the frst round. Suppose u 3 fres at u 4 n the frst round. Then u 4 has to fre at u 3 ; otherwse u 3 wll be an solated vertex after the frst round. But now u 5 wll be an solated vertex after the frst round. Hence u 3 fres at u 2, and smlarly u 5 fres at u 6. So at least one of u 3 and u 5 survves the frst round. Ths mples that u 4 has to survve the frst round as well. The result now follows by applyng Observatons 3 and 4 to the KO-successor of G. 3. NP-complete problems In ths secton, we consder two NP-complete problems that wll play a key role n our proof of Theorem 1. We refer to [7,10] for further detals. The frst problem concerns domnatng sets. A set S V s a domnatng set of a graph G = (V, E) f every vertex of G s n S or adjacent to a vertex n S. We wll make use of the followng NP-complete decson problem. DOMINATING SET (DS) Instance: A graph G = (V, E) and a postve nteger p. Queston: Does G have a domnatng set of cardnalty at most p? The second problem concerns hypergraph 2-colourngs. A hypergraph J = (Q, S) s a par of sets where Q = {q 1,..., q m } s the vertex set and S = {S 1,..., S n } s the set of hyperedges. Each member S j of S s a subset of Q. A 2-colourng of J = (Q, S) s a partton of Q nto sets B and W such that, for each S S, B S and W S. We wll also make use of the followng NP-complete decson problem. HYPERGRAPH 2-COLOURABILITY (H2C) Instance: A hypergraph J = (Q, S). Queston: Is there a 2-colourng of J = (Q, S)? Before we turn to our proofs of the complexty results n Secton 4, we need a few more defntons. The ncdence graph I of a hypergraph J = (Q, S) s a bpartte graph wth vertex set Q S where (q, S) forms an edge f and only f q S. Wth a hypergraph J = (Q, S) we can assocate another hypergraph J = (X, Z) called the trple of J; trples of hypergraphs wll play a crucal role n our NP-completeness proofs n Secton 4. It requres a lttle effort to defne the vertex set X and hyperedge set Z of the trple of J.
5 186 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) Fg. 1. Part of the ncdence graph of the trple of a hypergraph. Recall that Q = {q 1,..., q m } and S = {S 1,..., S n }. For 1 m, let l() be the number of hyperedges n S that contan q, let Q = {q 1,..., ql() } and let U = {u 1,..., ul() }. The unon of all such sets s the vertex set of J, that s m X = (Q U ). =1 Now the hyperedges. Let us frst defne the followng sets: for 1 m, for 1 k l(), let P k = {q k, uk }, for 1 m, for 1 k l() 1, let R k = T k for 1 m, let R l() = T l() = {u l(), q 1 }. = {u k, qk+1 }, and Let P = {P 1,..., Pl() }, R = {R 1,..., Rl() }, and T = {T 1,..., T l() }, and let P = m P, =1 R = m R, =1 T = m T. =1 For 1 j n, let us also defne a set S j. If n J, S j contans q, then n J, S j contans a vertex of Q. In partcular, f S j s the kth hyperedge that contans q n J, then S j contans qk. For example, f q 1 s n S 1, S 4 and S 7 (only) n J, then l(1) = 3 and n J there are vertces q 1 1, q2 1, q3 1 wth q1 1 S 1, q2 1 S 4, and q3 1 S 7. Let S = {S 1,..., S n }. The set of hyperedges for J s Z = S P R T. We denote the ncdence graph of the trple J of J by I. See Fg. 1 for an example that llustrates the case where q 1 belongs to S 1, S 4 and S 7. Proposton 7. The hypergraph J = (Q, S) has a 2-colourng B W f and only f ts trple J = (X, Z) has a 2-colourng B W such that for each 1 m ether Q B and U W, or Q W and U B. Proof. Suppose B W s a 2-colourng of J. Defne a partton B W of X as follows. If q s n B, then each q k s n B and each u k s n W. If q s n W, then each q k s n W and each u k s n B. Obvously, B W s a 2-colourng of J wth the desred property. Suppose we have a 2-colourng B W of J such that for each 1 m ether Q B and U W, or Q W and U B. Then let q B f and only f Q B, and let W = Q \ B. Clearly, f S j contans only elements from B (respectvely W ), then S j would contan only elements from B (respectvely W ). Hence B W s a 2-colourng of J.
6 4. Complexty classfcaton H. Broersma et al. / Theoretcal Computer Scence 393 (2008) We now have all the ngredents to prove our man complexty result. We repeat t here for convenence. Theorem 8. For k 2, PKO(k) s NP-complete even f nstances are restrcted to the class of bpartte graphs. Proof. It s clear that PKO(k) s n NP. The rest of the proof s n two cases. We gve separate proofs for the cases k = 2 and k 3. Case 1. k = 2. We use reducton from DS. Gven G = (V, E) and a postve nteger p V, we shall complete the proof by constructng a bpartte graph B such that pko(b) = 2 f and only f G has a domnatng set D wth D p. Let the vertex set of B be the dsjont unon of V = {v 1,..., v n }, V = {v 1,..., v n } and W = {w 1,..., w n p }. Let the edge set of B consst of v v, 1 n, v v j and v v j, for each edge v v j E, and v w h, 1 n, 1 h n p. Suppose that G has a domnatng set D = {v 1,..., v d } where d p. Note that every vertex n V s adjacent to a vertex of D n B. We shall descrbe a 2-round KO-reducton scheme for B. In the frst round for 1 n, v fres at v, for 1 j p, v j fres at v j, for p + 1 j n, v j fres at a vertex n D, and for 1 h n p, w h fres at a vertex n D. Thus each vertex n {v 1,..., v p } and V s elmnated n the frst round, and each vertex n V \ {v 1,..., v p } and W survves to round 2. As the survvng vertces nduce the balanced complete bpartte graph K n p,n p n B, t s clear that every survvng vertex can be elmnated n one further round. Now suppose that B has a 2-round KO-reducton scheme. Let D be the subset of V contanng vertces that are fred at n round 1. As every vertex n V fres at and so s adjacent to a vertex n D, D s a domnatng set n G (snce each vertex n V s joned only to copes of tself and ts neghbours). We complete the proof of Case 1 by showng that D p. Let V S = V \ D and V S V W be the sets of vertces that survve round 1. As round 2 s the fnal round, V S = V S. (1) As V W = 2n p and at most n vertces n V W are fred at n round 1, V S n p. Thus, by (1), V S n p. Therefore D = V V S n (n p) = p. Case 2. k 3. We use reducton from H2C. Let J = (Q, S) be an nstance of H2C. Let I be the ncdence graph of ts trple J = (X, Z). Recall that Z = S P R T. From I, we obtan another bpartte graph G by addng X + Z mutually vertex-dsjont paths and connectng each vertex of I wth one of these added paths as follows: For each vertex x n X, add a path H x = y x 1 yx 2 yx 3 and jon x to yx 1. For each vertex R n R, add a path H R = y R 1... y R 4 and jon R to y R 1. For each vertex T n T, add a path H T = y T 1... yt 4 and jon T to yt 1. For each vertex P n P, add a path H P = y P 1... y P 7 and jon P to the centrevertex y P 4. For each vertex S n S, add a path H S = y S 1... ys 7 and jon S to the centrevertex y S 4. Fg. 2 llustrates G.
7 188 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) Fg. 2. The graph G n Case 2. We complete the proof by showng that J s 2-colourable f and only f pko(g) k. Throughout the proof, G 1 and G 2 denote the graphs nduced by the survvng vertces of G after, respectvely, one and two rounds of a KO-reducton scheme. Suppose B W s a 2-colourng of J. By Proposton 7, J has a 2-colourng B W. We defne a three-round KO-reducton scheme for G, so we show that n ths case pko(g) 3 k. Round 1. Vertces of degree 1 and ther neghbours fre at each other. Each H P wth P P and each H S wth S S s a centred path of G, and the vertces fre as n Observaton 6. For each z R T, vertex y z 1 fres at yz 2 and y z 2 fres at yz 3. Each vertex n Z fres at one of ts neghbours n B. Each vertex x n X fres at ts neghbour y1 x n H x. Each y1 x wth x B fres at x. Each y1 x wth x W fres at y2 x. Thus every vertex n W and no vertex n B survves the frst round. Also every vertex n Z survves the frst round. After the frst round, each vertex z R T s adjacent to a vertex y z 1 of degree 1, and each vertex z S P s adjacent to a vertex y z 4 whose only other neghbour s a vertex yz 3 (or yz 5 ) of degree 1. Round 2. Because B W s a 2-colourng of J = (X, Z), every vertex n Z has a neghbour n W n G 1. For each S j S we choose one neghbour n W and let W be the set of selected vertces. Snce no two vertces n S have a common neghbour n X, W = n. The vertces n G 1 fre as follows. Vertces of degree 1 and ther neghbours fre at each other. Each vertex P P wth a neghbour n W \W fres at ths neghbour. Otherwse P fres at y4 P. Each x X fres at ts neghbour n P. Each S S fres at y4 S. Thus the vertex set of G 2 s W S. Round 3. Each S S and ts unque neghbour n W fre at each other, whch leaves us wth the null graph. Now we suppose that pko(g) k. We assume that a partcular KO-reducton scheme for G s gven and prove that J has a 2-colourng. We start wth the followng useful property. Clam 1. If a vertex of a set Q s knocked out n the frst round, then all the vertces of Q are knocked out n the frst round. Proof of Clam 1. Suppose that a vertex q k Q s knocked out n the frst round. We prove the clam by showng that q k+1 (wth q l()+1 = q 1 ) s also knocked out n the frst round. If q k Q s knocked out n the frst round, then, by Observaton 5, q k fres at y qk 1. Suppose qk+1 s not knocked out n the frst round. Observaton 6 mples that P k+1 must fre at u k+1, and P k must fre at ether q k or u k. If Pk fres at u k, then, by Observaton 5, uk fres at y qk Pk 1. Snce vertces n H must fre as n Observaton 6, ths means that G 1 contans a component somorphc to a path on three vertces. By Observaton 4, G 1 s not KO-reducble. Hence, P k fres at q k. For the same reason R k+1 or T k+1 cannot fre at u k Rk+1, and consequently they fre at y1 and y T k+1 1, respectvely. Due to Observaton 5 ths mples that y Rk+1 1 fres at y Rk+1 2, and y T k+1 1 fres at y T k+1 2. In G 1, T k and R k have exactly the same neghbours, namely u k and q k+1. If T k and R k fre at a dfferent neghbour n the second round, then due to Observaton 5 both wll be solated vertces n G 2. Suppose T k and R k fre at the same neghbour. Then n all possble schemes G 2 wll contan two vertces of degree 1 havng the same neghbour.
8 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) Observaton 4 mples that G 2 s not KO-reducble. We conclude that q k+1 must be knocked out n the frst round as well. Usng the same arguments, we get the followng clam. Clam 2. If a vertex n a set U s knocked out n the frst round, then all vertces n U are knocked out n the frst round. By Clams 1 and 2 we may defne a set B X as follows. All vertces of a set Q or U are n B f and only f the set s knocked out n the frst round. Let W = X\B. We need one more clam. Clam 3. For all 1 m, ether Q B and U W, or Q W and U B. Proof of Clam 3. Let 1 m. By Observaton 6, each vertex P k P must fre at ether q k or u k n the frst round. The prevous two clams mply that Q or U s knocked out n the frst round. Suppose both sets are knocked out n the frst round. Then, by Observaton 5, u 1 fres at y u1 1, and q1 fres at y q1 1 be knocked out n any round. The clam s proved.. Then, by Observaton 6, P1 wll not By Clam 3, all vertces n Z\S have one neghbour n B and one neghbour n W. Let S j be a vertex n S. By Observaton 6, S j fres at a neghbour n m =1 Q. By defnton, ths neghbour s n B. By Observatons 5 and 6, S j s knocked out by a neghbour n m =1 Q that s not knocked out n the frst round. By defnton, ths neghbour s n W. It s now clear that B W s a 2-colourng of J such that for each 1 m ether Q B and U W, or Q W and U B. Hence, by Proposton 7, the hypergraph J also has a 2-colourng. Ths completes the proof of Theorem 1. Theorem 1 has the followng two easy consequences. Corollary 9. The problem PKO s NP-complete, even f nstances are restrcted to the class of bpartte graphs. Proof. The problem PKO s clearly n NP. We use reducton from H2C. From an nstance J = (Q, S) we construct the graph G as n the proof of Theorem 1. We clam that J s 2-colourable f and only f G s KO-reducble. Suppose that J s 2-colourable. As we have seen n the proof of Theorem 1 ths mples that pko(g) 3. Hence G s KO-reducble. Suppose that G s KO-reducble. We copy the proof of Case 2 of Theorem 1. The second corollary of Theorem 1 nvolves the followng decson problem. EXACT PARALLEL KNOCK-OUT (k) (EPKO(k)) Instance: A graph G. Queston: Is pko(g) = k? Corollary 10. The problem EPKO(k) s polynomally solvable for k = 1 and s NP-complete for k 2, even f nstances are restrcted to the class of bpartte graphs. Proof. We already observed n Secton 1 that EPKO(1) s polynomally solvable. Ths mples that EPKO(2) s NPcomplete snce PKO(2) s NP-complete. For the case k 3 we make use of a famly of trees Y l wth pko(y l ) = l that have been constructed n [3]. For convenence, we recall the recursve defnton of two sequences Y 1, Y 2,... and Z 1, Z 2,... of rooted trees: The tree Y 1 conssts of a root wth one chld (Y 1 s a rooted P 2 ). The tree Z 1 conssts of a root wth one chld and one grandchld (Z 1 s a rooted P 3 ). For l 2, the tree Y l conssts of a root r and l dsjont subtrees. The frst l 2 of these subtrees are copes of the rooted trees Z 1,..., Z l 2 ; the last two of these subtrees are copes of Z l 1 ; r s adjacent to the roots of the l subtrees. For l 2, the tree Z l conssts of a root r and l subtrees. These subtrees are copes of the rooted trees Y 1,..., Y l ; r s adjacent to the roots of the l subtrees.
9 190 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) We add a dsjont copy of the tree Y k to the graph G constructed n the proof of Case 2 n Theorem 1. The new graph G has pko(g ) = k f and only f pko(g) k. Note that the sze of a tree Y k only depends on k and not on the sze of our nput graph G (so we do not need the exact descrpton of ths famly). We can even make the nstance graph connected by addng an edge between the neghbour of a leaf n Y k and the neghbour of a degree-one vertex n G. Note that H2C remans NP-complete for connected hypergraphs. Also note that by Observaton 3, n any KO-reducton scheme of the new graph a degree-one vertex and ts neghbour knock each other out n the frst round, so the added edges do not change the KO-reducblty propertes of the graph. 5. Bounded tree-wdth In ths secton we use monadc second-order logc; that s, that fragment of second-order logc where quantfed relaton symbols must have arty 1. For example, the followng sentence, whch expresses that a graph (whose edges are gven by the bnary relaton E) can be 3-coloured, s a sentence of monadc second-order logc: R W B { x ( (R(x) W (x) B(x)) (R(x) W (x)) (R(x) B(x)) (W (x) B(x)) ) ( x y E(x, y) )} ( (R(x) R(y)) (W (x) W (y)) (B(x) B(y))) (the quantfed unary relaton symbols are R, W and B, and should be read as sets of red, whte and blue vertces, respectvely). Thus, n partcular, there exst NP-complete problems that can be defned n monadc second-order logc. A semnal result of Courcelle [6] s that on any class of graphs of bounded tree-wdth, every problem defnable n monadc second-order logc can be solved n tme lnear n the number of vertces of the graph. Moreover, Courcelle s result holds not just when graphs are gven n terms of ther edge relaton, as n the example above, but also when the doman of a structure encodng a graph G conssts of the dsjont unon of the set of vertces and the set of edges, as well as unary relatons V and E to dstngush the vertces and the edges, respectvely, and also a bnary ncdence relaton I whch denotes when a partcular vertex s ncdent wth a partcular edge (thus, I V E). The reader s referred to [6] for more detals and also for the defnton of tree-wdth whch s not requred here. To prove Theorem 2, we need only prove the followng proposton. Proposton 11. For k 1, PKO(k) can be defned n monadc second-order logc. Proof. Recall that a parallel knock-out scheme for a graph G = (V, E) s a sequence of graphs G G 1 G 2 G r, where G r s the null graph. Let W 0 = V and, for 1 r, let W be the vertex set of G. If we can wrte a formula Φ(W, W +1 ) of monadc second-order logc that says there exsts a KO-selecton f on W such that the vertex set of the KO-successor s W +1, then we could prove the proposton wth the followng sentence Ω k whch s satsfed f and only f G s n PKO(k): W 0 W 1 W k { v(w 0 (v) V (v)) Φ(W 0, W 1 ) Φ(W 1, W 2 ) Φ(W k 1, W k ) ( v( W k (v) V (v)))} (Here and elsewhere we have presupposed that each W s a set of vertces; we could easly nclude addtonal clauses to check ths explctly.) The followng clam wll help us wrte Φ(W, W +1 ). Clam 4. There s a KO-selecton f on W such that W +1 s the vertex set of the KO-successor f and only f there s a partton V 1, V 2, V 3 of W and subsets E 1, E 2, E 3 of E such that
10 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) Fg. 3. A representaton of vertces frng. (a) for j = 1, 2, 3, each vertex n V j s ncdent wth exactly one edge of E j, ths edge jons t to a vertex n W \ V j, and ths accounts for every edge n E j (so V j = E j ). (b) W +1 W and, for j = 1, 2, 3, W +1 V j s the set of vertces n V j not ncdent wth edges n E j for any j j. We wll prove the clam later. Frst we use t to wrte Φ(W, W +1 ). The followng formula ψ(v 1, E 1, V 2, E 2, V 3, E 3, W ) checks that the sets V 1, V 2 and V 3 partton W, that the sets E 1, E 2, E 3 are edges n the graph, and that (a) s satsfed. v((v 1 (v) V 2 (v) V 3 (v)) W (v)) v( (V 1 (v) V 2 (v)) (V 1 (v) V 3 (v)) (V 2 (v) V 3 (v))) x((e 1 (x) E 2 (x) E 3 (x)) E(x)) x(e 1 (x) u v(v 1 (u) (V 2 (v) V 3 (v)) I (u, x) I (v, x))) x(e 2 (x) u v(v 2 (u) (V 1 (v) V 3 (v)) I (u, x) I (v, x))) x(e 3 (x) u v(v 3 (u) (V 1 (v) V 2 (v)) I (u, x) I (v, x))) v(v 1 (v)!x(i (v, x) E 1 (x))) v(v 2 (v)!x(i (v, x) E 2 (x))) v(v 3 (v)!x(i (v, x) E 3 (x))). (The semantcs of! s there exsts exactly one ; clearly, ths abbrevates a more complex though routne frst-order formula.) The followng formula checks that (b) s satsfed and s denoted χ(v 1, E 1, V 2, E 2, V 3, E 3, W, W +1 ). v(w +1 (v) (W (v) ((V 1 (v) x((e 2 (x) E 3 (x)) I (v, x))) (V 2 (v) x((e 1 (x) E 3 (x)) I (v, x))) (V 3 (v) x((e 1 (x) E 2 (x)) I (v, x)))))). And now we can wrte Φ(W, W +1 ): V 1 E 1 V 2 E 2 V 3 E 3 (ψ(v 1, E 1, V 2, E 2, V 3, E 3, W ) χ(v 1, E 1, V 2, E 2, V 3, E 3, W, W +1 )). It only remans to prove Clam 4. Suppose that we have sets V 1, V 2, V 3, E 1, E 2 and E 3 that satsfy the condtons of the clam. Then to defne the KO-selecton f, for j = 1, 2, 3, for each vertex v V j, let v fre at the unque neghbour joned to v by an edge n E j. It s easy to check that W +1 s the vertex set of the KO-successor. Now suppose that we have a KO-selecton f. Let H be the spannng subgraph of G wth edge set {v f (v) v W }. The frng can be represented as an orentaton of H: orent each edge from v to f (v) (some edges may be orented n both drectons). As each vertex has exactly one edge orented away from t, each component of the orented graph contans one drected cycle, of length at least 2, wth a pendant n-tree attached to each vertex of the cycle; see Fg. 3.
11 192 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) We fnd the sets V 1, V 2, V 3, E 1, E 2, E 3 ; the edge sets contan only edges of H. We may assume that H s connected (else we can fnd the sets componentwse). Let the vertces of the unque cycle n the orentaton be v 1,..., v c where the edges are v l v l+1, 1 l c 1, and v c v 1. So H contans vertces v 1,..., v c wth a pendant tree (possbly trval) attached to each. For 1 l c, let U l e be the set of vertces n the pendant tree attached to v l whose dstance from v l s even (but not zero), and let U l o be the vertces n the tree at odd dstance from v l. Let V 1 = Uo l Ue l {v l : l s even, l c}, l odd l even V 2 = Ue l Uo l {v l : l s odd, l c}, and l odd l even V 3 = {v c }, and, for = 1, 2, 3, let E contan v f (v) for each v V. It s clear that the sets we have chosen satsfy the condtons of the clam. Ths completes the proof of the clam and of the proposton. Theorem 2 follows from the proposton. And, notng that EPKO(k) s defned by the monadc second-order sentence Ω k Ω k 1, we have the followng result. Corollary 12. For k 1, EPKO(k) s solvable n lnear tme on any class of graphs wth bounded tree-wdth. In partcular, we obtan the followng result for trees, answerng an open queston n [3]. Corollary 13. For k 1, EPKO(k) s solvable n lnear tme for trees. Fnally, we note that to check whether a graph G s reducble t s suffcent to check whether pko(g) = k, for 1 k, where s the maxmum degree of G. Thus G s reducble f and only f the sentence Ω Ω 1 Ω 1 s satsfed. Ths gves us our last result of ths secton. Corollary 14. On any class of graphs wth bounded tree-wdth, PKO can be solved n polynomal tme. 6. Graphs wth a small parallel knock-out number As we noted n the ntroducton, graphs wth a [1, 2]-factor, or more partcularly wth a perfect matchng or wth a hamltonan cycle have parallel knock-out number 1. We start ths secton by studyng the related classes of factorcrtcal graphs and of 1-tough graphs. A graph G s sad to be factor-crtcal f G v has a perfect matchng for every vertex v of G. A graph G = (V, E) s called 1-tough f ω(g S) S for every nonempty subset S of V, where ω(g S) denotes the number of components of the graph G S. Clearly, every hamltonan graph s 1-tough and every factor-crtcal graph has a matchng leavng only one vertex unmatched. A natural queston s whether factorcrtcal graphs and 1-tough graphs have a small parallel knock-out number. The next results show that these graphs n fact have a [1, 2]-factor,.e., have parallel knock-out number 1 (unless they are trval,.e., contan only one vertex). We start wth factor-crtcal graphs. Theorem 15. Let G be a nontrval factor-crtcal graph and v V (G). Then G has a [1, 2]-factor consstng of an odd cycle C contanng v and a perfect matchng n G V (C). Proof. Let M be a perfect matchng n G v. If some neghbours of v are matched by M, we mmedately fnd the desred odd cycle (trangle) C and perfect matchng n G V (C), and we are done. Suppose ths s not the case. Then we take an arbtrary neghbour x of v, and note that there exsts a perfect matchng M n G x. Clearly x s matched to a vertex y v under M and v s matched to a vertex p under M. By our assumpton we may assume that p y; otherwse we fnd a trangle C wth the desred propertes. Snce both M and M saturate all vertces except for v and x, respectvely, there exsts an (M, M)-alternatng path P between v and y begnnng and endng wth an edge of M. Now P together wth the edges between x and v and x and y forms an odd cycle C, and the remanng edges of M form a perfect matchng n G V (C).
12 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) The above result also mples that nontrval 1-tough graphs on an odd number of vertces have a [1, 2]-factor. In order to prove ths, we need the followng well-known result of Tutte [11]. Let ω o (G) denote the number of odd components of a graph G,.e., the number of components contanng an odd number of vertces. Theorem 16 ([11]). A graph G has a perfect matchng f and only f ω o (G S) S for all S V (G). Ths theorem has the followng consequence. Corollary 17. If G s a 1-tough graph on an odd number of vertces, then G s factor-crtcal. Proof. Suppose G s 1-tough on an odd number of vertces, but not factor-crtcal. Then there exsts a vertex v V (G) such that G = G v has no perfect matchng. Thus by Theorem 16 there exsts a set X V (G ) wth ω o (G X ) = X k, for some nteger k 0. Settng X = X {v}, and lettng ω e denote the number of even components, we have ω(g X) = ω o (G X) + ω e (G X) = ω o (G X ) + ω e (G X ) = X k + ω e (G X ) = X + k + ω e (G X ) X 1. Snce G s 1-tough, k = 0 and ω e (G X ) = 0; otherwse ω(g X) > X 1, a contradcton. Let H 1,..., H X +1 denote the odd components of G X. Then V (G) = 1 + X + X +1 =1 V (H ) = X +1 =1 ( V (H ) + 1). Snce all V (H ) are odd, V (G) s even, a contradcton. Corollary 18. Every nontrval 1-tough graph has a [1, 2]-factor. Proof. Consder a nontrval 1-tough graph G on n vertces. If n s odd the result follows by combnng Theorem 15 and Corollary 17. If n s even G has a perfect matchng by Theorem 16. We now turn to regular graphs and almost regular bpartte graphs. Frst we note that the trck ntroduced n Secton 1 mmedately mples that every r-regular graph G wth r 1 has a [1, 2]-factor,.e., pko(g) = 1. Proposton 19. Every r-regular graph G wth r 1 has a [1, 2]-factor. Proof. Let G be an r-regular graph wth r 1 and wth V (G) = {v 1, v 2,..., v n }. Defne a bpartte graph G wth vertex set {u 1, u 2,..., u n, w 1, w 2,..., w n } n whch u w j E(G ) and u j w E(G ) f and only f v v j E(G). Then G s an r-regular bpartte graph and has a perfect matchng (See, e.g., [2] Exercse 5.2.3(a)). Ths matchng corresponds to a [1, 2]-factor n G. The above result also mmedately mples the followng statement for graphs that contan a k-factor,.e., a spannng k-regular subgraph. Corollary 20. Every nontrval graph wth a k-factor has a [1, 2]-factor. Our complexty results for bpartte graphs motvated us to consder bpartte graphs that are almost regular n the followng sense. A bpartte graph G s called (r, s)-regular f all vertces n one class of the bpartton have degree r and all other vertces have degree s. By Proposton 19 any (r, r)-regular bpartte graph G wth r 1 has pko(g) = 1, and one easly checks that any (1, s)-regular bpartte graph G wth s 2 has pko(g) =. Wth the next result we characterze all reducble (2, s)-regular bpartte graphs, notng that (2, s)-regular graphs wth s 2, 3 are not reducble. Let G be a (2, 3)-regular bpartte graph, and let L denote the vertces wth degree 2 (left vertces) and R the vertces wth degree 3 (rght vertces). Then E(G) = 2 L = 3 R, so R = 2k and L = 3k for some postve nteger k. We call a subset A of R wth k vertces that has the whole set L as ts neghbourhood a k-star cover of G. Clearly ths mples that all vertces of A have mutually dsjont neghbours n L. We wll also need the noton of an f -factor and a result due to Ore. If f s an nteger-valued functon on the set V (G) such that 0 f (v) d G (v) for each vertex
13 194 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) v V (G), then a spannng subgraph F of G s called an f -factor of G f d F (v) = f (v) for each vertex v V (G). The followng theorem of Ore (see, e.g., [1], Theorem 7.2.2) characterzes bpartte graphs wth an f -factor. In ths theorem E(U, y) denotes the set of edges between a vertex set U and a vertex y, and f (U) = v U f (v). Theorem 21. Let G be a bpartte graph wth bpartton (L, R). Then G has an f -factor f and only f f (L) = f (R) and for any set U of R: f (U) y L mn( f (y), E(U, y) ). Wth the adopton of the above conventons and result we can prove the followng result for (2, 3)-regular bpartte graphs. Theorem 22. Let G be a (2, 3)-regular bpartte graph. Then G s reducble f and only f pko(g) = 2 f and only f G has a k-star cover. Moreover, we can determne pko(g) n polynomal tme. Proof. Let G be a (2, 3)-regular bpartte graph, and let L denote the vertces wth degree 2 and R the vertces wth degree 3. If G s reducble, then pko(g) = 2: t cannot be 1 snce L > R, and t cannot be larger than 2 snce n every round the degree of the vertces decreases by at least 1 and the vertces n R cannot elmnate each other snce R s an ndependent set. Ths clearly proves the frst equvalence of the statement. For the same reasons, f pko(g) = 2, then n the second round the remanng vertces of L have degree 1, and those vertces and the remanng vertces of R elmnate each other along a perfect matchng M. We now show that M = k. Frst of all, M k snce R = 2k and hence the vertces of R can elmnate at most 2k vertces of L n the frst round; secondly, M k snce otherwse all 3k vertces n L elmnate fewer than k from R n round 1, contradctng that every vertex of R has degree 3 n G. Snce the remanng k vertces of R are saved n round 1, all 3k left vertces elmnate together only R k = k rght vertces. Snce all vertces n R have degree 3, ths scheme corresponds to a k-star cover of G. Conversely, suppose G has a k-star cover A n R. Then B = R \ A s also a k-star cover of G. Now we set f (x) = 1 for all x A L, f (x) = 2 for all x B. Then f (L) = 3k and f (R) = k + 2k = 3k, so f (L) = f (R). For a set U R, f (U) = U A + 2 U B 3 max( U A, U B ). Snce A s a k-star cover, all neghbours of the vertces of U A n L are dstnct, and the same holds for B and U B. Any neghbour y L of each of the vertces from U A or U B clearly has mn( f (y), E(U, y) ) 1 snce f (y) = 1 and y s a neghbour of a vertex of U. So y L mn( f (y), E(U, y) ) 3 max( U A, U B ). Usng Theorem 21, we conclude that G has an f -factor. Consder the followng KO-scheme for G: n round 1, all vertces n L fre at A, whle vertces n A fre va matchng edges of the f -factor and vertces n B fre along one of the edges of the f -factor. After round 1, all remanng vertces at the rght sde are precsely the set B. Because of the f -factor n G and the frng n the frst round, the vertces of B form a perfect matchng M wth the remanng vertces n L. We use M to elmnate all remanng vertces n the second round. Ths completes the proof of the second equvalence of the statement. Determnng whether G has a k-star cover s a problem that can be solved n polynomal tme. Snce both A and B must be k-star covers, one can start by puttng one arbtrary vertex v of R n A, and then puttng all vertces of R that have a common neghbour wth v n B, and so on, untl all vertces have been allocated or a conflct occurs. The cases of reducble (r, s)-regular bpartte graphs for other values of r and s do not seem to admt a smlar characterzaton. We leave them as nterestng open problems. 7. Conclusons In ths paper we have studed the computatonal complexty of problems related to the parallel knock-out number pko(g) of a graph G. We have shown that determnng whether pko(g) = 1 s polynomally solvable, whereas determnng whether pko(g) k (or pko(g) = k) s NP-complete for any fxed k 2 that s not part of the nput, even when restrcted to the class of bpartte graphs. We also showed that the latter problems restrcted to graphs wth bounded tree-wdth are solvable n lnear tme, by formulatng them n monadc second-order logc. Moreover, we studed some specal graph classes wth small parallel knock-out numbers. An nterestng open problem s the computatonal complexty of both the decson problems when restrcted to planar graphs. Snce outer-planar graphs have bounded tree-wdth, both problems can be solved n lnear tme when restrcted to outer-planar graphs. Snce 4-connected planar graphs are hamltonan, pko(g) = 1 for a 4-connected planar graph G. From a result n [3] we can easly deduce that pko(g) 20 log n for any reducble planar graph G on n vertces.
14 H. Broersma et al. / Theoretcal Computer Scence 393 (2008) References [1] A.S. Asratan, T.M.J. Denley, R. Häggkvst, Bpartte Graphs and ther Applcatons, Cambrdge Unversty Press, Cambrdge, [2] J.A. Bondy, U.S.R. Murty, Graph Theory wth Applcatons, Macmllan, London, (Elsever, New York). [3] H.J. Broersma, F.V. Fomn, R. Královč, G.J. Woegnger, Elmnatng graphs by means of parallel knock-out schemes, Dscrete Appled Mathematcs 155 (2) (2007) [4] H.J. Broersma, F. Göbel, k-traversable graphs, Ars Combnatora 29A (1990) [5] H.J. Broersma, M. Johnson, D. Paulusma, Upper bounds and algorthms for parallel knock-out numbers, n: Proceedngs of SIROCCO 2007: 14th Internatonal Colloquum on Structural Informaton and Communcaton Complexty, n: Lecture Notes n Computer Scence, vol. 4474, 2007, pp [6] B. Courcelle, The monadc second-order logc of graphs. I. Recognzable sets of fnte graphs, Informaton and Computaton 85 (1990) [7] M.R. Garey, D.S. Johnson, Computers and Intractablty: A Gude to the Theory of NP-Completeness, Freeman, San Francsco, [8] B. Jackson, N.C. Wormald, k-walks, Australasan Journal of Combnatorcs 2 (1990) [9] D.E. Lampert, P.J. Slater, Parallel knockouts n the complete graph, Amercan Mathematcal Monthly 105 (1998) [10] L. Lovász, Coverng and colorng of hypergraphs, n: Proceedngs of the 4th Southeastern Conference on Combnatorcs, Graph Theory, and Computng, Utltas Mathematca, 1973, [11] W.T. Tutte, The factorzaton of lnear graphs, Journal of London Mathematcal Socety 22 (1947)
The computational complexity of the parallel knock-out problem
The computatonal complexty of the parallel knock-out problem Hajo Broersma 1 Matthew Johnson 1 Danël Paulusma 1 Ian A. Stewart 1 1 Department of Computer Scence Durham Unversty South Road, Durham, DH1
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationEdge Isoperimetric Inequalities
November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationThe L(2, 1)-Labeling on -Product of Graphs
Annals of Pure and Appled Mathematcs Vol 0, No, 05, 9-39 ISSN: 79-087X (P, 79-0888(onlne Publshed on 7 Aprl 05 wwwresearchmathscorg Annals of The L(, -Labelng on -Product of Graphs P Pradhan and Kamesh
More informationDiscrete Mathematics
Dscrete Mathematcs 30 (00) 48 488 Contents lsts avalable at ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc The number of C 3 -free vertces on 3-partte tournaments Ana Paulna
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationn ). This is tight for all admissible values of t, k and n. k t + + n t
MAXIMIZING THE NUMBER OF NONNEGATIVE SUBSETS NOGA ALON, HAROUT AYDINIAN, AND HAO HUANG Abstract. Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what
More informationEvery planar graph is 4-colourable a proof without computer
Peter Dörre Department of Informatcs and Natural Scences Fachhochschule Südwestfalen (Unversty of Appled Scences) Frauenstuhlweg 31, D-58644 Iserlohn, Germany Emal: doerre(at)fh-swf.de Mathematcs Subject
More informationSelf-complementing permutations of k-uniform hypergraphs
Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationTHE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS
Dscussones Mathematcae Graph Theory 27 (2007) 401 407 THE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS Guantao Chen Department of Mathematcs and Statstcs Georga State Unversty,
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationDiscrete Mathematics. Laplacian spectral characterization of some graphs obtained by product operation
Dscrete Mathematcs 31 (01) 1591 1595 Contents lsts avalable at ScVerse ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc Laplacan spectral characterzaton of some graphs obtaned
More informationarxiv: v2 [cs.ds] 1 Feb 2017
Polynomal-tme Algorthms for the Subset Feedback Vertex Set Problem on Interval Graphs and Permutaton Graphs Chars Papadopoulos Spyrdon Tzmas arxv:170104634v2 [csds] 1 Feb 2017 Abstract Gven a vertex-weghted
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationA Simple Research of Divisor Graphs
The 29th Workshop on Combnatoral Mathematcs and Computaton Theory A Smple Research o Dvsor Graphs Yu-png Tsao General Educaton Center Chna Unversty o Technology Tape Tawan yp-tsao@cuteedutw Tape Tawan
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationA new construction of 3-separable matrices via an improved decoding of Macula s construction
Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More informationModulo Magic Labeling in Digraphs
Gen. Math. Notes, Vol. 7, No., August, 03, pp. 5- ISSN 9-784; Copyrght ICSRS Publcaton, 03 www.-csrs.org Avalable free onlne at http://www.geman.n Modulo Magc Labelng n Dgraphs L. Shobana and J. Baskar
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationPAijpam.eu SOME NEW SUM PERFECT SQUARE GRAPHS S.G. Sonchhatra 1, G.V. Ghodasara 2
Internatonal Journal of Pure and Appled Mathematcs Volume 113 No. 3 2017, 489-499 ISSN: 1311-8080 (prnted verson); ISSN: 1314-3395 (on-lne verson) url: http://www.jpam.eu do: 10.12732/jpam.v1133.11 PAjpam.eu
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationSpectral Graph Theory and its Applications September 16, Lecture 5
Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph
More informationZ 4p - Magic labeling for some special graphs
Internatonal Journal of Mathematcs and Soft Computng Vol., No. (0, 6-70. ISSN Prnt : 49-8 Z 4p - Magc labelng for some specal graphs ISSN Onlne: 9-55 V.L. Stella Arputha Mary Department of Mathematcs,
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationarxiv: v1 [cs.gt] 14 Mar 2019
Stable Roommates wth Narcssstc, Sngle-Peaked, and Sngle-Crossng Preferences Robert Bredereck 1, Jehua Chen 2, Ugo Paavo Fnnendahl 1, and Rolf Nedermeer 1 arxv:1903.05975v1 [cs.gt] 14 Mar 2019 1 TU Berln,
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationUNIQUE FACTORIZATION OF COMPOSITIVE HEREDITARY GRAPH PROPERTIES
UNIQUE FACTORIZATION OF COMPOSITIVE HEREDITARY GRAPH PROPERTIES IZAK BROERE AND EWA DRGAS-BURCHARDT Abstract. A graph property s any class of graphs that s closed under somorphsms. A graph property P s
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationMixed-integer vertex covers on bipartite graphs
Mxed-nteger vertex covers on bpartte graphs Mchele Confort, Bert Gerards, Gacomo Zambell November, 2006 Abstract Let A be the edge-node ncdence matrx of a bpartte graph G = (U, V ; E), I be a subset the
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationk(k 1)(k 2)(p 2) 6(p d.
BLOCK-TRANSITIVE 3-DESIGNS WITH AFFINE AUTOMORPHISM GROUP Greg Gamble Let X = (Z p d where p s an odd prme and d N, and let B X, B = k. Then t was shown by Praeger that the set B = {B g g AGL d (p} s the
More information(k,?)-sandwich Problems: why not ask for special kinds of bread?
Couto-Fara-Klen-Prott-Noguera MC 2014/4/9 19:04 page 17 #1 Matemátca Contemporânea, Vol. 42, 17 26 c?2014, Socedade Braslera de Matemátca (k,?)-sandwch Problems: why not ask for specal knds of bread? F.
More informationFinding Primitive Roots Pseudo-Deterministically
Electronc Colloquum on Computatonal Complexty, Report No 207 (205) Fndng Prmtve Roots Pseudo-Determnstcally Ofer Grossman December 22, 205 Abstract Pseudo-determnstc algorthms are randomzed search algorthms
More informationDistribution of subgraphs of random regular graphs
Dstrbuton of subgraphs of random regular graphs Zhcheng Gao Faculty of Busness Admnstraton Unversty of Macau Macau Chna zcgao@umac.mo N. C. Wormald Department of Combnatorcs and Optmzaton Unversty of Waterloo
More informationThe Second Eigenvalue of Planar Graphs
Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationTheoretical Computer Science
Theoretcal Computer Scence 412 (2011) 1263 1274 Contents lsts avalable at ScenceDrect Theoretcal Computer Scence journal homepage: www.elsever.com/locate/tcs Popular matchngs wth varable tem copes Telkepall
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More informationComputing Correlated Equilibria in Multi-Player Games
Computng Correlated Equlbra n Mult-Player Games Chrstos H. Papadmtrou Presented by Zhanxang Huang December 7th, 2005 1 The Author Dr. Chrstos H. Papadmtrou CS professor at UC Berkley (taught at Harvard,
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationIntroductory Cardinality Theory Alan Kaylor Cline
Introductory Cardnalty Theory lan Kaylor Clne lthough by name the theory of set cardnalty may seem to be an offshoot of combnatorcs, the central nterest s actually nfnte sets. Combnatorcs deals wth fnte
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationGenericity of Critical Types
Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f
More informationMATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1
MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε
More informationCCO Commun. Comb. Optim.
Communcatons n Combnatorcs and Optmzaton Vol. 2 No. 2, 2017 pp.87-98 DOI: 10.22049/CCO.2017.13630 CCO Commun. Comb. Optm. Reformulated F-ndex of graph operatons Hamdeh Aram 1 and Nasrn Dehgard 2 1 Department
More informationBeyond Zudilin s Conjectured q-analog of Schmidt s problem
Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationOn quasiperfect numbers
Notes on Number Theory and Dscrete Mathematcs Prnt ISSN 1310 5132, Onlne ISSN 2367 8275 Vol. 23, 2017, No. 3, 73 78 On quasperfect numbers V. Sva Rama Prasad 1 and C. Suntha 2 1 Nalla Malla Reddy Engneerng
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationLecture 20: Lift and Project, SDP Duality. Today we will study the Lift and Project method. Then we will prove the SDP duality theorem.
prnceton u. sp 02 cos 598B: algorthms and complexty Lecture 20: Lft and Project, SDP Dualty Lecturer: Sanjeev Arora Scrbe:Yury Makarychev Today we wll study the Lft and Project method. Then we wll prove
More informationarxiv: v3 [cs.dm] 7 Jul 2012
Perfect matchng n -unform hypergraphs wth large vertex degree arxv:1101.580v [cs.dm] 7 Jul 01 Imdadullah Khan Department of Computer Scence College of Computng and Informaton Systems Umm Al-Qura Unversty
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationSimultaneous Optimization of Berth Allocation, Quay Crane Assignment and Quay Crane Scheduling Problems in Container Terminals
Smultaneous Optmzaton of Berth Allocaton, Quay Crane Assgnment and Quay Crane Schedulng Problems n Contaner Termnals Necat Aras, Yavuz Türkoğulları, Z. Caner Taşkın, Kuban Altınel Abstract In ths work,
More informationA CLASS OF RECURSIVE SETS. Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA
A CLASS OF RECURSIVE SETS Florentn Smarandache Unversty of New Mexco 200 College Road Gallup, NM 87301, USA E-mal: smarand@unmedu In ths artcle one bulds a class of recursve sets, one establshes propertes
More informationREGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction
REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997
More informationU.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan 2/21/2008. Notes for Lecture 8
U.C. Berkeley CS278: Computatonal Complexty Handout N8 Professor Luca Trevsan 2/21/2008 Notes for Lecture 8 1 Undrected Connectvty In the undrected s t connectvty problem (abbrevated ST-UCONN) we are gven
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationCaps and Colouring Steiner Triple Systems
Desgns, Codes and Cryptography, 13, 51 55 (1998) c 1998 Kluwer Academc Publshers, Boston. Manufactured n The Netherlands. Caps and Colourng Stener Trple Systems AIDEN BRUEN* Department of Mathematcs, Unversty
More informationStanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011
Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More informationLecture Space-Bounded Derandomization
Notes on Complexty Theory Last updated: October, 2008 Jonathan Katz Lecture Space-Bounded Derandomzaton 1 Space-Bounded Derandomzaton We now dscuss derandomzaton of space-bounded algorthms. Here non-trval
More informationRandom Walks on Digraphs
Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected
More informationEuropean Journal of Combinatorics
European Journal of Combnatorcs 0 (009) 480 489 Contents lsts avalable at ScenceDrect European Journal of Combnatorcs journal homepage: www.elsever.com/locate/ejc Tlngs n Lee metrc P. Horak 1 Unversty
More informationarxiv: v1 [math.co] 7 Apr 2015
Ranbow connecton n some dgraphs Jesús Alva-Samos 1 Juan José Montellano-Ballesteros Abstract arxv:1504.0171v1 [math.co] 7 Apr 015 An edge-coloured graph G s ranbow connected f any two vertces are connected
More informationDistance Three Labelings of Trees
Dstance Three Labelngs of Trees Jří Fala Petr A. Golovach Jan Kratochvíl Bernard Ldcký Danël Paulusma Abstract An L(2, 1, 1)-labelng of a graph G assgns nonnegatve ntegers to the vertces of G n such a
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationSubset Topological Spaces and Kakutani s Theorem
MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationTHERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q.
THERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q. IAN KIMING We shall prove the followng result from [2]: Theorem 1. (Bllng-Mahler, 1940, cf. [2]) An ellptc curve defned over Q does not have a
More informationA combinatorial problem associated with nonograms
A combnatoral problem assocated wth nonograms Jessca Benton Ron Snow Nolan Wallach March 21, 2005 1 Introducton. Ths work was motvated by a queston posed by the second named author to the frst named author
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationSpectral graph theory: Applications of Courant-Fischer
Spectral graph theory: Applcatons of Courant-Fscher Steve Butler September 2006 Abstract In ths second talk we wll ntroduce the Raylegh quotent and the Courant- Fscher Theorem and gve some applcatons for
More informationOn C 0 multi-contractions having a regular dilation
SUDIA MAHEMAICA 170 (3) (2005) On C 0 mult-contractons havng a regular dlaton by Dan Popovc (mşoara) Abstract. Commutng mult-contractons of class C 0 and havng a regular sometrc dlaton are studed. We prove
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationDeterminants Containing Powers of Generalized Fibonacci Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More information