Fractional Zero Forcing via Three-color Forcing Games

Transcription

1 Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that is caied out on the -blowup of a gaph is intoduced and used to define the factional positive semidefinite focing numbe. Popeties of the gaph blowup when coloed with a factional positive semidefinite focing set ae examined and used to define a thee-colo focing game that diectly computes the factional positive semidefinite focing numbe of a gaph. We also pesent a thee-colo intepetation of the skew zeo focing game. The teatment of factional positive semidefinite focing numbe is paalleled to develop a factional paamete based on the standad zeo focing pocess and it is shown that this paamete is exactly the skew zeo focing numbe. The theecolo appoach and an algoithm ae used to chaacteize gaphs whose skew zeo focing numbe equals zeo. Key wods. zeo focing, factional, positive semidefinite, gaph AMS subject classifications. 05C72, 05C50, 05C57, 05C Intoduction This pape studies factional vesions (in the spiit of [9]) of the standad and positive semidefinite zeo focing numbes and intoduces thee-colo focing games to compute these paametes Zeo focing games The zeo focing pocess was intoduced independently in [1] as a method of focing zeos in a null vecto of a matix descibed by a gaph in ode to uppe bound the nullity of the matix and in [4] fo contol of quantum systems. The oiginal pocess has since spawned numeous vaiants. In this section, we intoduce zeo focing games and the teminology used theein. Depatment of Mathematics, Iowa State Univesity, Ames, IA 50011, USA (LHogben@iastate.edu) and Ameican Institute of Mathematics, 600 E. Bokaw Rd., San Jose, CA 95112, USA (hogben@aimath.og). Depatment of Mathematics, Iowa State Univesity, Ames, IA 50011, USA (kpalmow@iastate.edu). Division of Mathematical Sciences, Nanyang Technological Univesity, SPMS-MAS-03-01, 21 Nanyang Link, Singapoe (dobeson@ntu.edu.sg). Depatment of Mathematics, Iowa State Univesity, Ames, IA 50011, USA (myoung@iastate.edu). 1

2 Abstactly, a focing game is a type of coloing game that is played on a simple gaph G. Fist, a taget colo, typically blue o dak blue, is designated. Each vetex of the gaph is then coloed the taget colo, white, o possibly some othe colo (in pio wok, only white and the taget colo have been used). A focing ule is chosen: this is a ule that descibes the conditions unde which some vetex can cause anothe vetex to change to the taget colo. If vetex u causes a neighboing vetex w to change colo, we say that u foces w and wite u w. The focing ule is epeatedly applied until no moe foces can be pefomed, at which point the game ends; the coloing at the end is called the final coloing. An odeed list of the foces pefomed is efeed to as a chonological list of foces. Note that thee is usually some choice as to which foces ae pefomed, as well as the ode in which these foces occu. As such, a single focing set may geneate many diffeent chonological lists of foces; howeve, the final coloing is unique fo all of the games discussed heein. If the gaph is totally coloed with the taget colo at the end of the game, then we say that G has been foced. The goal of the game is to foce G. If this is possible, then the initial set of non-white vetices is called a focing set. The (standad) zeo focing game uses only the colos blue (the taget colo) and white. The (standad) zeo focing ule is as follows: If w is the only white neighbo of a blue vetex u, then u can foce w. A (standad) zeo focing set is an initial set of blue vetices that can foce G using this ule. The (standad) zeo focing numbe of G, denoted Z(G), is the minimum cadinality of a zeo focing set fo G. We pesent an illustative example. a d f a d f a d f c b e g (a) Initial focing set c b e g (b) Fist thee foces c b e g (c) Final foce Figue 1: Standad zeo focing game example Example 1.1. Let G be as in Figue 1 and choose the initial set of blue vetices B = {a, f, g} (Figue 1a). Since each vetex in B has only one white neighbo, we ae able to pefom the foces a c, f d, and g e; Figue 1b shows the state of the system afte these fist foces ae pefomed. Afte this, the only white vetex emaining in the gaph is b, which is then foced by c (Figue 1c). Thus we have foced G and conclude that B is a (standad) zeo focing set; it is left as an execise to veify that B is a minimum (standad) zeo focing set, so Z(G) = B = 3. Fom this point fowad, we will omit the wod standad when efeing to the standad zeo focing game, its focing ule, o zeo focing sets wheneve thee is no isk of ambiguity. The positive semidefinite zeo focing game is a modification of the zeo focing game used to foce zeos in a null vecto of a positive semidefinite matix descibed by a gaph [3]. Like the zeo 2

3 focing game, positive semidefinite zeo focing uses only the colos blue (taget) and white. The positive semidefinite zeo focing ule is the same as the standad zeo focing ule, except that this ule also featues a disconnect ule: Remove all blue vetices fom the gaph, leaving a set of connected components. To each connected component (of white vetices) in tun, add the blue vetices, the edges among the blue vetices, and any edges between the blue vetices and that component, and pefom foces via the standad ule: If w is the only white neighbo of a blue vetex u in this induced subgaph, then u can foce w. It is not assumed that disconnection occus; if thee is only one component, then we simply foce via the standad focing ule. If disconnection does occu, then afte the foce the gaph is eassembled pio to applying the ule again. As one would expect, a positive semidefinite zeo focing set is an initial set of blue vetices that can foce G using this ule, and the positive semidefinite zeo focing numbe of G, denoted Z + (G), is the minimum cadinality of a positive semidefinite zeo focing set fo G. As in the standad zeo focing case, we examine an illustative example. a d d f a d f a d f c d c d c b e g (a) Initial focing set c b e g (b) Connected components c c b e g (c) Focing in each component a d f a d f c b e g (d) Reassembled gaph c b e g (e) Final foce Figue 2: Positive semidefinite zeo focing game example Example 1.2. Let G be as in Figue 2 and choose the initial set of blue vetices B = {c, d} (Figue 2a). This is clealy not a standad zeo focing set, since no initial foce can be made using the standad zeo focing ule; howeve, the positive semidefinite zeo focing game allows us to use the disconnect ule, and this example eveals its powe. Applying the disconnect ule yields the connected components shown in Figue 2b. B is then connected to each component and one foce is pefomed in each component (Figue 2c). Afte focing, the gaph is eassembled (Figue 2d). 3

4 The final foce in the pocess, e g, does not equie the disconnect ule (Figue 2e). As befoe, we wee able to foce G, so the initial set B is a positive semidefinite zeo focing set; it is left as an execise to veify that B is also minimum and Z + (G) = 2. The skew zeo focing game, anothe vaiant on zeo focing that uses the colos white and blue (taget), was fist consideed in [8] to foce zeos in a null vecto of a skew symmetic matix descibed by a gaph. The skew zeo focing ule is as follows: If w is the only white neighbo of any vetex u, then u can foce w. Skew zeo focing emoves the standad equiement that the focing vetex u be blue; as a esult, skew zeo focing allows white vetex focing, i.e., a white vetex is allowed to foce its only white neighbo. A skew zeo focing set is an initial set of blue vetices that can foce G using this ule, and the skew zeo focing numbe of G, denoted Z (G), is the minimum cadinality of a skew zeo focing set fo G. We etun fo a final time to ou illustative example. a d f a d f a d f c b e g (a) Initial focing set c b e g (b) Fist thee foces Figue 3: Skew zeo focing game example c b e g (c) Final thee foces Example 1.3. Let G be as in Figue 3 and choose the initial blue vetex B = {a} (Figue 3a). The vetex a is able to pefom a standad foce on its neighbo c, and vetices f and g ae able to pefom white vetex foces on thei neighbos d and e, espectively (Figue 3b). At this point, the standad foces c b, d f, and e g can be pefomed, which foces G (Figue 3c). B is thus a skew zeo focing set; as befoe, it is an execise to show that B is minimum and Z (G) = Motivation and method This pape focuses on factional vesions of the standad and positive semidefinite zeo focing numbes. We fist pesent the constuction of factional chomatic numbe found in [9] as an example of the method used to define a factional gaph paamete. A pope coloing of a gaph G is an assignment of colos to the vetices of G such that adjacent vetices eceive diffeent colos. The chomatic numbe of G, denoted χ(g), is the least numbe of colos equied to popely colo G. We can genealize a pope coloing of G using c colos to a pope -fold coloing with c colos, o a c:-coloing: fom a total of c colos, we assign colos to each vetex of G such that adjacent vetices eceive disjoint sets of colos. The -fold chomatic numbe of G, denoted χ (G), is the smallest value of c such that G has a c:-coloing; we emphasize that to compute χ (G) we fix and minimize the value of c. The factional chomatic numbe of G is then defined as { } χ (G) χ f (G) = inf. N 4

5 The inteested eade is efeed to [9] fo an in-depth teatment of factional chomatic numbe, as well as othe factional gaph paametes. Fo this pape, defining an -fold vesion of a gaph paamete and then defining the factional paamete as the infimum of the atios of the -fold paamete to ae key ideas. Suppose that G is a simple gaph on n vetices with V (G) = {1, 2,..., n}. We say that a matix A C n n -fits G if, afte patitioning A as a block n n matix, block A ii = I fo each i and fo all i, j with i j, block A ij = 0 if and only if ij / E(G) [7]. While thee may be many such matices fo a given gaph, the following esult shows that cetain stuctue can be assumed. Poposition 1.4. Suppose that A C n n -fits a gaph G on n vetices. We can constuct a unitay matix U such that U AU -fits G and if ij E(G), then evey enty of block (U AU) ij is nonzeo. Poof. Assume that V (G) = {1, 2,..., n} and patition A = [A ij ] as an n n block matix with A ij C. By definition, we have A ii = I fo each i [1 : n] and fo i, j [1 : n] with i j we have A ij = 0 if and only if ij / E(G). Fo each i [1 : n], let U i C be a andom unitay matix with U i and U j chosen independently if i j. Define U = blockdiag(u 1,..., U n ) and let C = U AU. Patitioning C confomally with A, we have C ij = Ui A iju j. Notice that C ii = Ui I U i = I and fo i j if ij / E(G), then C ij = Ui 0 U j = 0. Suppose ij E(G) and conside the poduct A ij U j ; note that A ij 0. Since U j is andom, with high pobability no column of U j lies in ke A ij, so no column of A ij U j is a zeo vecto. Let z be any column of A ij U j (so with high pobability z 0) and conside (Ui z) k. If (Ui z) k = 0, then z is othogonal to the k th column of U i. Since U i is a andom unitay matix, with high pobability this does not happen. We conclude that if ij E(G), then with high pobability no enty of C ij is zeo. Thus C -fits G and has the desied stuctue. Let G be a gaph and choose N. The -blowup of G is the gaph G () constucted by eplacing each vetex of u V (G) with an independent set of vetices, denoted R u, and eplacing each edge uw E(G) by the edges of a complete bipatite gaph on patite sets R u and R w. 1 We call the set R u a cluste. Note that V (G () ) = u V (G) R u and if uw E(G) then evey vetex of R u is adjacent to evey vetex of R w in G (). Suppose that A C n n is positive semidefinite and -fits a gaph G on n vetices with V (G) = {1, 2,..., n}. By Poposition 1.4, without loss of geneality we can assume that if ij / E(G), then block A ij has no zeo enties. Conside the gaph of such a matix A, namely, the simple gaph with vetex set {1, 2,..., n} and with an edge between vetices k and l if k l and the enty in ow k and column l of A is nonzeo. Since A ii = I, the vetices of G will map to independent sets (clustes) of size ; let R i denote the cluste associated with vetex i V (G). Since each enty of A ij is nonzeo, evey vetex in R i will be adjacent to evey vetex in R j, and vice vesa. Hence the gaph of A is exactly G (), the -blowup of G. 1 Given gaphs G and H, the lexicogaphic poduct of G with H, denoted G L H, is the gaph with V (G L H) = V (G) V (H) and (g, h)(i, j) E(G L H) if gi E(G) o if g = i and hj E(H). We can also define the -blowup of G as G () = G L K, whee K denotes the empty gaph on vetices. 5

6 The positive semidefinite zeo focing numbe of a gaph is an uppe bound on the maximum positive semidefinite nullity of the gaph, which equals the ode of the gaph minus its minimum positive semidefinite ank [3, 5]. The authos of [7] define an -fold analogue of minimum positive semidefinite ank and use this new paamete to define factional minimum positive semidefinite ank. A key element of this teatment is that the -fold minimum positive semidefinite ank of a gaph can be expessed as the ank of a positive semidefinite matix that -fits the gaph [7, Theoem 3.9]. Ou pevious discussion allows us to assume that the gaph of such a matix is G (). As mentioned in Section 1.1, playing the positive semidefinite zeo focing game can be intepeted as focing zeos in a null vecto of a positive semidefinite matix whose gaph is G, hence the connection to maximum positive semidefinite nullity and minimum positive semidefinite ank. Since the -fold minimum positive semidefinite ank is defined in tems of matices that -fit the oiginal gaph, an -fold analogue of positive semidefinite zeo focing numbe would natually be associated with a game played on the gaph of a positive semidefinite matix that -fits G. To this end, ou -fold focing paametes will be defined in tems of focing games played on G () ; while the intepetation of focing zeos in a null vecto is no longe valid, the spiit of the pocess emains Definitions and notation Thoughout this pape, all gaphs ae simple. We use G to denote the ode of a gaph G, i.e., G = V (G). If G is a gaph and S V (G), then G[S] denotes the subgaph of G induced by S, namely, the gaph with V (G[S]) = S and E(G[S]) = {uv E(G) : u, v S}. We use G S as shothand fo the induced subgaph G[V (G) \ S]. The neighbohood of a vetex u V (G), denoted N(u), is the set of vetices adjacent to u. The degee of a vetex u is the numbe of neighbos of u, i.e., N(u). A leaf is a vetex of degee one. We use δ(g) to denote the minimum of the degees of the vetices of G. If S and T ae disjoint sets, then S T denotes the disjoint union of the sets. Note that S T = S T ; we use the notation to emphasize that the sets ae disjoint. Thoughout, B will be used to denote a set of blue vetices associated with a two-colo focing game. We emphasize that in a two-colo focing game the taget colo is blue. Fo thee-colo focing games, we use two non-white colos: dak blue, which is ou taget colo, and light blue. B will be used to denote a set of coloed vetices associated with a thee-colo focing game. Given such a set B, we let D be the set of dak blue vetices and L be the set of light blue vetices. Since D L =, we have B = D L. While B is a set, we will abuse notation and wite B = (D, L) to emphasize the decomposition of B into its component sets Contibution and oganization of the pape 159 In Section 2 we intoduce and examine the factional positive semidefinite focing numbe of a gaph. 160 An -fold extension of the positive semidefinite zeo focing numbe, based on gaph blowups, is 161 intoduced and used to define the factional positive semidefinite focing numbe of a gaph G, denoted Z + f (G). We also intoduce a thee-colo focing game played on G called the factional 163 positive semidefinite focing game and pove a main esult of that section (cf. Theoem 2.21): 6

7 Theoem. Fo any gaph G, Z f (G) is the minimum numbe of dak blue vetices in a (thee-colo) 165 factional positive semidefinite focing set fo G This esult allows us to detemine the factional positive semidefinite focing numbe of a gaph by playing the factional positive semidefinite focing game, as opposed to computation via the -fold appoach. We pove numeous esults petaining to factional positive semidefinite focing numbe and the stuctue of optimal factional positive semidefinite focing sets and apply these esults to compute the factional positive semidefinite focing numbe fo some common gaph families. We also pove that any gaph has an odinay (two-colo) minimum positive semidefinite zeo focing set such that the fist foce in the focing pocess can be done without using the disconnect ule. In Section 3 we intoduce a thee-colo focing game that is equivalent to the skew zeo focing game. The thee-colo appoach is used to pove numeous esults petaining to skew zeo focing. We define an -fold analogue of the (standad) zeo focing game and using this to define the factional focing numbe of a gaph, denoted Z f (G). A main esult of that section shows that skew zeo focing numbe and factional zeo focing numbe of a gaph ae the same (cf. Theoem 3.21): Theoem. Fo any gaph G, Z f (G) = Z (G). We conclude the section by intoducing an algoithm that is used to chaacteize gaphs that satisfy Z (G) = Factional positive semidefinite focing In this section, we define an -fold analogue of the positive semidefinite zeo focing game and use this to define the -fold and factional positive semidefinite focing numbes of a gaph G. We investigate stuctual popeties of -fold positive semidefinite focing sets and use these popeties to develop a simple thee-colo game to diectly compute the factional positive semidefinite focing numbe of a gaph. Popeties of the factional positive semidefinite focing numbe ae also investigated The -fold positive semidefinite focing game and factional positive semidefinite focing numbe Let G be a gaph and fo some N conside the following -fold positive semidefinite focing game, which is a two-colo focing game played on G (). As in any focing game, we initially colo some set B V (G () ) blue and then ty to foce G () though epeated application of the following -fold positive semidefinite focing ule: Definition 2.1 (-fold positive semidefinite focing ule). Let B t denote the set of blue vetices of G () at some step t of the -fold positive semidefinite focing pocess 2 and let W 1,..., W h denote 2 We caution the eade that a chonological list of foces is not a popagating pocess and B t hee has diffeent meaning than that used in the study of popagation. 7

8 the sets of vetices of the connected components of G () B t. If u B t and N(u) W i, then u can foce N(u) W i, i.e., all white neighbos of u in G () [B t W i ] can be simultaneously coloed blue The -fold positive semidefinite focing game can be thought of as a genealization of the positive semidefinite zeo focing game: instead of focing one white neighbo in a component afte applying the disconnect ule, a vetex foces up to white neighbos in a component. This is a positive semidefinite analog of the -focing pocess descibed in [2], but we apply this pocess only to the blowup of the gaph. 206 If G () can be foced, then the initial set of blue vetices is called an -fold positive semidefinite 207 (PSD) focing set fo G. An -fold PSD focing set B is minimum if thee is no -fold PSD focing 208 set of smalle cadinality than B. The cadinality of a minimum -fold PSD focing set is called the -fold positive semidefinite focing numbe of G and is denoted Z [](G). We define the factional 210 positive semidefinite focing numbe of a gaph G as { Z + Z + [] f (G) = inf (G) } 211. N 212 Note that G (1) = G and a 1-fold PSD focing set is exactly a positive semidefinite zeo focing 213 set. Any positive semidefinite zeo focing set B can be conveted into an -fold PSD focing set (fo 214 2) by the following ule: if u B, then colo evey vetex in R u V (G () ) blue. This ceates an -fold PSD focing set that contains Z + (G) blue vetices, so Z + [] (G) Z+ (G) = Z + [1] (G). 216 We conclude that { Z + Z + [] f (G) = inf } { Z + [] = inf } 217. N Global intepetation of -fold positive semidefinite focing Suppose that we ae playing the -fold positive semidefinite focing game on G (), whee 2. So fa, we have viewed the game fom a local pespective while geneally ignoing the global stuctue of the blowup, namely, clustes joined by edges. Shifting to a global view gives insight into the mechanics of the focing game. In this section, we assume that 2. Thee specific types of cluste ae of paticula inteest. An All cluste is a cluste in which all vetices ae coloed blue. A One cluste is a cluste in which exactly one vetex is coloed blue and the est ae coloed white. A None cluste is a cluste in which all vetices ae coloed white. We define a All-One-None (minimum) -fold positive semidefinite focing set B fo a gaph G to be a (minimum) -fold PSD focing set in which each cluste of G () is eithe an All, One, o None cluste when G () is coloed with B. Fo the sake of bevity, we will heeafte shoten All-One-None to AON. We say that a cluste R u is foced into when any vetex in R u is foced. Once a cluste changes fom a non-all to an All cluste, we say that the cluste has been foced. Obsevation 2.2. Any cluste that is foced into becomes an All cluste afte the focing opeation, so focing into a cluste and focing the cluste ae equivalent. 8

9 Remak 2.3. At some stage of the -fold positive semidefinite focing pocess using a paticula chonological list of foces, let B t denote the set of blue vetices in G (). Assume that R u B t fo some u V (G). Suppose that the next foce in the pocess is done by x R u, so x has at most white neighbos. Since R u B t, thee exists at least one white vetex w R u. Because x and w have the same neighbos and w is white, all white neighbos of x ae connected though w and lie in the same connected component. Hence, afte x foces, all neighbos of evey vetex in R u must be blue, so without loss of geneality R u can be foced in the next step of the focing pocess. This emak yields a new definition. Definition 2.4. If at any stage of the -fold positive semidefinite focing pocess a vetex in any patially-filled cluste pefoms a foce, then that cluste can itself be foced at the next focing step. We efe to this pocess as backfocing. Remak 2.3 assets that equiing backfocing does not affect whethe o not a set is an -fold PSD focing set, so we will always assume that backfocing is used when pefoming the -fold positive semidefinite focing pocess. As we will see, this assumption is quite poweful. Definition 2.5. Let R u1, R u2,..., R um be patially-filled clustes (i.e., no cluste is an All o a None) in G () that togethe contain p + q blue vetices fo some 0 p < m and 0 q <. We define the pocess of consolidation as follows: use p of the blue vetices to convet R u1,..., R up into All clustes and move the emaining q blue vetices into R up+1. Consolidation allows us to liteally consolidate a goup of blue vetices spead among many clustes into the fewest numbe of clustes possible. Ou goal fo the emainde of this section is to use these tools and definitions to develop an equivalent chaacteization of the -fold positive semidefinite focing game that elies only upon a paticula type of AON -fold PSD focing set. Remak 2.6. Suppose that 3. If an -fold focing set B ceates a global AON stuctue in G (), then fom a global pespective exactly one cluste is foced at each step of the focing pocess. This is because the vetex that pefoms the foce can only foce into One o None clustes, and if this vetex wee adjacent to moe than one of these (in any combination), then it would have moe than white neighbos and could not actually pefom a foce. The case when = 2 is slightly diffeent. In this case, it is possible fo a vetex to foce two One clustes at the same focing step (cf. Example 2.11 below). Evey 2-fold PSD focing set is automatically an AON set, so we cannot claim that if G () has a global AON stuctue, then exactly one cluste will be foced at the next focing step. Howeve, Theoem 2.7 uses consolidation to show that even though evey AON PSD focing set need not have this popety, thee always exist an AON minimum PSD focing set and focing pocess that do. Theoem 2.7. Let G be a gaph and suppose 2. Then thee exists an AON minimum -fold PSD focing set fo G. Fo all 3, exactly one cluste of G () will be foced at each step of any focing pocess that begins with any such set. Fo = 2, thee exists a focing pocess fo the set constucted such that exactly one cluste of G () is foced at each step. 9

10 Poof. We fist conside the case whee 3. Let B be a minimum -fold PSD focing set fo G and assume that B is not AON. Wite a chonological list of the foces pefomed using the focing set B, assuming the use of backfocing, and let B t, t 0, denote the set of blue vetices afte step t of this focing pocess, whee B 0 = B. Suppose that a vetex x R u pefoms a foce at step l 1 of the focing pocess and R u B l 1. By Obsevation 2.2, R u was not foced into at any step pio to step l. Since we assume backfocing and R u contains at least one white vetex, R u was not used to foce any othe cluste pio to step l, and R u will be foced in step l + 1. Thus if R u is not a One cluste, we can uncolo evey blue vetex in R u except fo x without changing the ability of x to foce o the ability of R u to be backfoced at step l + 1; since R u is not involved in any foces pio to step l, we can make this change in the oiginal set B and obtain a focing set with fewe blue vetices, contadicting the assumption that B was a minimum focing set. Thus evey cluste in a minimum -fold PSD focing set that is not an All cluste and contains a vetex that pefoms a foce must be a One cluste. Now, suppose that at step l 1 we have x W (R u1 R u2 R um ) fo some m 2, whee each R uj contains at least one white vetex. Since x is pefoming a foce, it has at most white neighbos in the component containing m j=1 R u j, so thee ae at least (m 1) blue vetices in m j=1 R u j. By Obsevation 2.2, each cluste R uj is an All cluste afte step l, and no R uj was foced into pio to step l. Since we assume backfocing and each of the R uj clustes contains at least one white vetex, none of the R uj clustes contains a vetex that was used to foce at a step pio to step l. Analogous to Remak 2.3, emoving blue vetices fom any of the R uj will not affect the application of the disconnect popety, as each R uj contains at least one white vetex. Similaly, adding blue vetices to convet an R uj into an All cluste may make available additional disconnects (which we do not use), but these would not affect any pevious foces. Theefoe, we can consolidate the (at least (m 1)) blue vetices in m j=1 R u j without affecting the ability to pefom any pevious foce. Without loss of geneality, suppose that R u1,..., R um 1 become All clustes afte the consolidation and any emaining blue vetices ae left in R um. Afte consolidation, the new foce at step l will be x R um ; afte this point, the state of the system is the same as it would have been had we not consolidated (i.e., evey R uj is an All cluste), so futue foces ae unaffected by consolidation. Futhemoe, afte consolidation, exactly one cluste (R um ) is foced at step l. Since the consolidation pocess does not affect any of the foces befoe o afte the foce at step l, we ae fee to pefom the consolidation on the oiginal set B to obtain a new minimum -fold PSD focing set B and the sequence of vetices that pefom foces emains unchanged. Note that since B is minimum, R um must necessaily be a None cluste: if not, then we could emove the blue vetices in R um and obtain a valid focing set with fewe blue vetices, contadicting that B is minimum. By epeated application of the consolidation pocess, we ae able to convet evey non-one cluste into an All cluste o a None cluste. By Remak 2.6, any AON focing pocess fo 3 must necessaily consist of focing only one cluste at each step, which poves the claim fo 3. Now, suppose that = 2. Evey minimum 2-fold PSD focing set fo G is automatically an AON set. Suppose that, at step l 1 of the focing pocess, moe than one cluste must be foced. Since any vetex can foce at most 2 of its neighbos, it must be the case that two One clustes ae 10

11 foced at this step. Fo the easons descibed in the 3 case, we can consolidate these two One clustes into one All cluste and one None cluste without affecting any pevious o futue foces; afte this consolidation, only one cluste (the None) is foced at step l. Thus we can modify ou oiginal minimum focing set (as befoe) and the esult follows fo the = 2 case (using the focing pocess to which consolidation was applied). We call the type of AON minimum -fold PSD focing set guaanteed to exist by Theoem 2.7 an optimal AON -fold PSD focing set. We emphasize that an optimal AON -fold PSD focing set is minimum by definition, and given an optimal AON -fold PSD focing set thee is a coesponding focing pocess in which exactly one cluste is foced at each step. Futhe, the set of blue vetices at each step of the focing pocess associated with an optimal AON -fold PSD focing set will always ceate a global AON stuctue in G (). Suppose that B is an AON -fold PSD focing set fo a gaph G and colo G () with B. We use a(b) to denote the numbe of All clustes in G () and l(b) to denote the numbe of One clustes in G (). This implies that B = a(b) + l(b). This new teminology yields a coollay to Theoem 2.7. Coollay 2.8. Fo evey gaph G and 2, if B is any optimal AON -fold PSD focing set fo G, then Z + [](G) = B = a(b) + l(b). Definition 2.9. Let, s 2 with s and suppose that B is an AON -fold PSD focing set fo G. Copy the AON stuctue of G () when coloed with B onto G (s) to ceate a new AON set of blue vetices of cadinality s a(b) + l(b). This pocess is called eplication. Remak It is possible that eplicating a 2-fold PSD focing set B fo G onto G (s) fo some s > 2 may not yield a valid focing set; this would occu when, at some step of the focing pocess on G (2), two One clustes ae foced simultaneously (see Example 2.11). Howeve, if B is an optimal AON 2-fold PSD focing set, then Theoem 2.7 guaantees that thee is a focing pocess in which exactly one foce occus at each step, so eplication will yield a valid focing set. Thus if B is obtained by eplicating an optimal AON -fold PSD focing set onto G (s) fo some, s 2 with s, then B is an AON s-fold PSD focing set fo G and the same focing pocess used on G () will wok with B. As we see in Example 2.12, howeve, B may not be minimum and hence not optimal. (a) (Minimum) AON 2-fold PSD focing set (b) Optimal AON 2-fold PSD focing set Figue 4: AON 2-fold PSD focing sets fo K Example Conside the (minimum) 2-fold PSD focing sets fo K 3 shown in Figue 4. Fo simplicity, the edges in the figue epesent the complete bipatite gaphs between the clustes at thei endpoints. The fist focing step in Figue 4a would consist of focing two of the One clustes simultaneously. This set is no longe a focing set when eplicated onto K (s) 3 fo s 3, as each of 11

12 the blue vetices will have too many white neighbos to pefom a foce. The optimal AON PSD focing set shown in Figue 4b, howeve, can be eplicated successfully, as only one cluste must be foced at any step of the focing pocess. (a) Gaph G (b) = 2 (c) = 3 Figue 5: Optimal AON -fold PSD focing sets 351 Example Suppose that we have the complete bipatite gaph K 5,2 and let G be the gaph 352 fomed by attaching one leaf to each of the vetices in the patite set containing five vetices (Figue 353 5a). Conside the optimal AON -fold PSD focing sets fo G shown in Figue 5. When = 2 (Figue 5b), the (unique) minimum PSD focing set ceates two All clustes, so Z + [2](G) = 4. When = 3 (Figue 5c), the (unique) minimum PSD focing set ceates five One clustes, so Z + [3](G) = In this case, eplicating eithe set onto the othe blowup will geneate a focing set that is not 357 minimum, hence not optimal We have shown that the -fold PSD focing numbe of a gaph can be computed using an optimal AON -fold PSD focing set. We now pove futhe popeties of AON -fold PSD focing sets and use these esults to povide an altenate definition of the factional PSD focing numbe. Lemma Let G be a gaph on n vetices and choose n. Fo any AON -fold PSD focing set B thee exists an AON -fold PSD focing set B with B B, a( B) = a(b), and l( B) < n. Poof. If a(b) = n, then clealy l(b) < n, so choose B = B. Now, assume that a(b) < n. If n 3, then only one cluste is foced at each step of any focing pocess. If n 2, then since a(b) < n we must have = n = 2 and again one cluste is foced at each focing step. Fo any, if the fist cluste foced with some focing pocess is completely white, then l(b) < n and we let B = B. If not, let B be the set obtained by eplacing the fist cluste foced with a None cluste. Lemma Let G be a gaph on n vetices and fix n. Let B be an optimal AON -fold PSD focing set fo G and let B be an AON -fold PSD focing set fo G. Then a(b) a(b ). Poof. By Lemma 2.13, we can assume without loss of geneality that l(b ) < n. Since B is optimal, it is minimum, so a(b) + l(b) = B B = a(b ) + l(b ). Dividing though by and manipulating this inequality yields a(b) a(b ) l(b ) l(b) < n 1. 12

13 Since a(b) a(b ) is an intege, we must have a(b) a(b ) 0, which poves the claim. Coollay Let G be a gaph on n vetices and fix n. If B and B ae optimal AON -fold PSD focing sets fo G, then a(b) = a(b ). Thus fo a fixed lage enough, evey optimal AON -fold PSD focing set fo G must contain the same numbe of All clustes (and, consequently, One clustes). Of paticula inteest is the case = n = G. We define a + (G) to be the unique numbe of All clustes ceated in G (n) by any optimal AON n-fold PSD focing set fo G, and define l + (G) to be the unique numbe of One clustes ceated in this manne. Ou next esult shows that once n, inceasing will not change the numbe of All clustes ceated by an optimal AON -fold PSD focing set (i.e., the numbe will emain the constant a + (G)). Poposition Let G be a gaph on n vetices. Fo all n, if B is an optimal AON -fold PSD focing set fo G, then a(b) = a + (G). Poof. Let B be the AON n-fold PSD focing set fomed by eplicating B onto G (n). By Lemma 2.14, a + (G) a( B) = a(b). Similaly, let B be the AON -fold PSD focing set fomed by eplicating any optimal AON n-fold PSD focing set onto G (). By Lemma 2.14, a(b) a(b ) = a + (G), and thus equality holds. Poposition 2.16 yields an elegant desciption of the -fold positive semidefinite focing numbe fo n, which we state as a coollay. Coollay Let G be a gaph on n vetices. Fo all n, Z + [] (G) = a+ (G) + l + (G). Additionally, Z + [] lim (G) = a + (G). Befoe we can pove the final esult of this section, which ties the factional positive semidefinite focing numbe into the machiney developed in this section, we equie one final utility esult. Lemma Let G be a gaph on n vetices and choose 2. Then fo any optimal AON -fold PSD focing set B, B a + (G). Poof. Fist, suppose that 2 < n. Let B be the AON n-fold PSD focing set obtained by eplicating B onto G (n). Then a(b) = a( B) and l(b) = l( B), so B l(b) = a(b) + = a( B) l( B) + a( B) l( B) + n = B n. Let B be any optimal AON n-fold PSD focing set fo G. Since B is optimal, it is minimum, hence B B. Theefoe, 406 B B n B n = a+ (G) + l+ (G) a + (G), n 13

14 which poves the claim fo < n. If n, then Poposition 2.16 shows that B = a + (G)+l + (G) and the conclusion follows. We conclude this section with an altenate chaacteization of factional positive semidefinite focing numbe. Theoem Fo evey gaph G, Z + f (G) = a+ (G). 413 Poof. Recall that Z + f = inf 2 { Z + [] (G) }. By Coollay 2.17, Z + f (G) a+ (G) Let B be an optimal AON -fold PSD focing set fo G. Then by Coollay 2.8 and Lemma 2.18, Z + [] (G) = B a+ (G), and thus equality holds. This esult shows that the factional positive semidefinite focing numbe of a gaph is always a nonnegative intege hence, it is factional in name (and constuction) only Thee-colo intepetation of factional positive semidefinite focing Motivated by the AON intepetation of the -fold positive semidefinite focing game, we conside a thee-colo focing game that allows us to compute the factional positive semidefinite focing numbe fo any gaph without playing the -fold game. Let G be a gaph and conside the following factional positive semidefinite focing game, which is a thee-colo focing game that uses the colos dak blue (taget), light blue, and white. Assign to each vetex of G one of these colos and let B = (D, L), whee D denotes the set of dak blue vetices and L denotes the set of light blue vetices. 3 We epeatedly apply the following factional positive semidefinite focing ule: Definition 2.20 (factional positive semidefinite focing ule). Let B t = (D t, L t ) denote the set of coloed vetices of a gaph G at some step of the factional positive semidefinite focing pocess and let W 1,..., W h denote the sets of vetices of the connected components of G D t. If u (D t (L t W i )) and w W i is the only light blue o white neighbo of u in G[D t W i ], then u can foce w, i.e., w can be coloed dak blue. Loosely speaking, we apply the disconnect ule fom positive semidefinite zeo focing using the dak blue vetices of G, and then in each econstucted component any dak o light blue vetex can foce its only light blue o white neighbo. As usual, the goal of this focing game is to choose the initial set B in such a way that by epeated application of this ule the entie gaph can be 3 Recall that this is equivalent to witing B = D L; see also Section

15 foced (i.e., tuned dak blue). If G can be foced, then we say that the initial set B is a factional positive semidefinite (PSD) focing set fo G. The (thee-colo) factional positive semidefinite focing numbe of G, denoted Ẑ+ f (G), is then defined as Ẑ + f (G) = min { D : (D, L) is a factional PSD focing set fo G, fo some L}. 442 We say that a factional PSD focing set B = (D, L) fo G is optimal if D = Ẑ+ f (G) and no factional PSD focing set fo G with D = Ẑ+ (G) has fewe than L light blue vetices. We use ˆl f (G) to denote the numbe of light blue vetices in any optimal factional PSD focing set fo G, i.e., ˆl + (G) = L. The pocess of backfocing descibed fo the -fold positive semidefinite focing game applies to the factional positive semidefinite focing game, albeit with a thee-colo modification. Afte a light blue vetex u pefoms a foce, all of its neighbos must necessaily be dak blue, and so we can backfoce u at the next focing step. As in the -fold case, backfocing is a poweful technique: once a light blue vetex is able to foce its only non-dak-blue neighbo, it can itself then be foced in the next step. The obsevant eade will notice that we have defined factional positive semidefinite focing numbe twice: hee, and in Section 2.1. The final esult of this section shows that this is not an eo: the paamete Z f, defined via an -fold two-colo game, is equal to the paamete Ẑ+ f, which 455 is defined via a thee-colo game Theoem Fo any gaph G on n vetices, Z + f (G) = Ẑ+ f (G). 458 Poof. Let B be an optimal AON n-fold PSD focing set fo G. By Theoem 2.19, we have a(b) = a + (G) = Z f (G). Let B = (D, L) be an optimal factional PSD focing set fo G, and note that 460 optimality implies that D = Ẑ+ f (G) Colo G (n) with B. Colo G with B = ( D, L), defined as follows: let D = {u : R u is an All cluste in G (n) } and let L = {u : R u is a One cluste in G (n) }. Since B is an optimal AON n-fold PSD focing set, exactly one cluste is foced at each step of the focing pocess using B, and G (n) can be foced. Futhe, backfocing is applied to One clustes in G (n), and One clustes coespond to light blue vetices, to which backfocing can also be applied. Theefoe, the focing pocess (fom a global viewpoint) used on G (n) can be used to foce G, so B is a factional PSD focing set fo G and Ẑ+ f (G) D = a(b) = Z + f (G) Now, colo G with B. Colo G (n) as follows, and let B be the set of blue vetices: if u D, then let R u be an All cluste, and if u L, then let R u be a One cluste. Since B is a focing set, B is an AON n-fold PSD focing set fo G (with essentially the same focing pocess). By Lemma 2.14, we have a(b) a( B), so Z + f (G) = a(b) a( B) = D = Ẑ+ f (G) and thus equality holds. Coollay Fo any gaph G, l + (G) = ˆl + (G). As a consequence of these esults, the Ẑ+ f and ˆl + notations will be suppessed in favo of the simple Z + f and l+. 15

16 In contast to the pocess of computing the values of factional vesions of geneal gaph pa- 476 ametes, computing the factional positive semidefinite focing numbe of a gaph does not equie any explicit knowledge of the -fold analog. If knowledge of Z + f is all that is of inteest, one can 478 bypass the -fold game entiely and opt to play the factional positive semidefinite focing game 479 instead Results fo factional positive semidefinite focing numbe The factional positive semidefinite focing game allows us to easily pove many inteesting popeties of the factional positive semidefinite focing numbe Remak Any isolated vetex in G must be coloed dak blue. (G) 1. Z + f Thus if δ(g) = 0, then Obsevation If G has connected components {G i } m 1, then Z+ f (G) = m 1 Z+ f (G i) and l + (G) = m 1 l+ (G i ). In light of this obsevation, we ae able to focus on connected gaphs. Remak Z + f (G) Z+ (G) Z(G). The fist inequality holds because any positive semidefinite zeo focing set fo G can be thought of as a factional PSD focing set fo G with Z + (G) dak blue vetices, and the second inequality is well-known (cf. [5]). Poposition Let G be a gaph and let B = (D, L) be a factional PSD focing set fo G. Then Z + (G) B = D + L. Futhe, Z + (G) Z + f (G) + l+ (G). Poof. B = D L is a positive semidefinite zeo focing set fo G, so Z + (G) B = D + L. The second claim follows by choosing B to be optimal. A natual question is whethe Z + (G) = Z + f (G) + l+ (G) in geneal. By taking a minimum positive semidefinite zeo focing set fo G and changing some vetices to light blue, it may be possible to obtain an optimal factional PSD focing set fo G. While this technique does wok fo many natual examples, the esult does not hold fo evey gaph, as the next example shows. Figue 6: Gaph fom Example 2.27 with p = 5, q = 2 16

17 Example Let G be the genealization of the gaph fom Example 2.12, whee instead of K 5,2 we use K p,q with patite sets P and Q satisfying P = p > q = Q 2. By coloing each of these leaves light blue, we can foce all of P, and using the disconnect ule we can subsequently backfoce the leaves and foce all of Q. Thus Z + f (G) = 0 and l+ (G) = p, but it is known that Z + (G) = q < 0 + p = Z + f (G) + l+ (G). The key to this example is that the set B = L is a minimal positive semidefinite zeo focing set fo G, but it is not a minimum positive semidefinite zeo focing set. Remak If B = (D, L) is an optimal factional PSD focing set fo a connected gaph G, then any vetex that is coloed light blue must pefom a foce befoe it is itself foced; if not, then that vetex can be coloed white to obtain a factional PSD focing set with the same numbe of dak blue vetices and fewe light blue vetices, contadicting the optimality of B. Additionally, no two light blue vetices in an optimal factional PSD focing set can be adjacent, as one would have to foce the othe befoe the othe has pefomed a foce. Theefoe, L is an independent set in G, so l + (G) α(g). We now pesent two esults fom positive semidefinite zeo focing. Remak Let G be a gaph. At some step t of the positive semidefinite zeo focing pocess, let B t denote the set of blue vetices and W 1,..., W h denote the sets of vetices of the connected components of G B t. Then by Remak in [11] we may select any i and pefom the next foce in G[B t W i ]. Lemma 2.30 ([10], Lemma 2.1.1). Let G be a gaph and let B be a positive semidefinite zeo focing set of G. If v B is the vetex that pefoms the fist foce, v w, whee w is a white neighbo of v, then (B \ {v}) {w} is a positive semidefinite zeo focing set of G. The following esult is a thee-colo vesion of Lemma The poof is simila to the poof of the two-colo vesion found in [10] and is omitted. Lemma Let G be a gaph and let B = (D, L) be a factional PSD focing set fo G. Suppose that the fist foce, v w, is pefomed by some v D on some w / L. Let D = (D \ {v}) {w}. Then B = ( D, L) is also a factional PSD focing set fo G. Notice that if B is an optimal factional PSD focing set fo G, then the fist vetex foced in G must necessaily be white. This obsevation lets us apply Lemma 2.31 to any optimal factional PSD focing set, povided that the fist foce is done by a dak blue vetex. Theoem If G is a gaph with at least one edge, then G has an optimal factional PSD focing set with which the fist foce can be pefomed by a light blue vetex. Poof. The esult is tivially tue fo any optimal factional PSD focing set with which the fist foce can be pefomed by a light blue vetex. Note that if the fist foce with an optimal set can be done without using the disconnect ule, then this foce must be done by a light blue vetex, else the set is not optimal. 17

18 Suppose fo the sake of contadiction that G does not have an optimal factional PSD focing set with which the fist foce can be pefomed by a light blue vetex. By the pevious agument, the disconnect ule must be applied to pefom the fist foce with any optimal factional PSD focing set. Let B = (D, L) be an optimal factional PSD focing set such that W 1 is minimum, whee W 1, W 2,..., W h ae the sets of vetices of the connected components of G D and W 1 W 2 W h. By Remak 2.29 we can assume that the fist vetex foced lies in W 1. Let v w be the fist foce, whee v D by assumption and w W 1. By Lemma 2.31, the set B = ( D, L) with D = (D \ {v}) {w} is also an optimal factional PSD focing set fo G. Since w must be the only non-dak-blue neighbo of v in W 1, it must be the case that v joins a component othe than W 1 in G D; futhe, in G D, the component W 1 will not contain the vetex w, and may split into multiple smalle components. If W 1 {w}, then this agument shows that thee must be a component with fewe than W 1 vetices in G D, which contadicts the choice of B; thus we must have W 1 = {w}. Howeve, the fist foce in G using B can theefoe be chosen as w v, which can be done without applying the disconnect ule; by the comments above, w can thus be light blue, contadicting optimality of B. We conclude that G must have an optimal factional PSD focing set with which the fist foce can be pefomed by a light blue vetex. Theoem 2.32 yields a lowe bound on Z + f (G) as a coollay. Coollay Fo any gaph G, δ(g) 1 Z + f (G). Poof. The esult is tivial fo δ(g) 1. If δ(g) 2, then G has an edge, so by Theoem 2.32 thee exists some optimal factional PSD focing set B = (D, L) such that the fist foce in G can be done by some u L. Remak 2.28 assets that u has no light blue neighbos, and all white neighbos of u must be in the same component of G D. Since u can foce, all but one of its neighbos must be dak blue. Thus D N(u) 1 δ(g) 1. An additional coollay to Theoem 2.32 gives a lowe bound on l + (G) in the case whee G has at least one edge. Coollay If G is a gaph with at least one edge, then l + (G) 1. The following esult is a two-colo analogue of Theoem 2.32 that applies to the positive semidefinite focing game. The poof is simila to that of Theoem 2.32 and is omitted. Theoem If G is a gaph with at least one edge, then thee exists a minimum positive semidefinite zeo focing set fo G such the fist foce can be done without using the disconnect ule. With Theoem 2.35, we can obtain an uppe bound on Z + f (G). Coollay Fo any gaph G with at least one edge, Z + f (G) Z+ (G) 1. 18

19 Poof. Theoem 2.35 ensues that thee is some minimum positive semidefinite zeo focing set B such that the fist foce using B can be done without using the disconnect ule. If B is obtained by coloing the vetex that pefoms this fist foce light blue and all of the othe vetices in B dak blue, then B is a factional PSD focing set with Z + (G) 1 dak blue vetices Factional positive semidefinite focing numbes fo gaph families In this section, we detemine the factional PSD focing numbes fo common gaph families, illustating the utility of some of the esults in Section 2.4. Example Let n 2 and let V (K n ) = {v 1, v 2,..., v n }. Note that Z + (K n ) = n 1 (cf. [5, Example ]). Applying Coollaies 2.33 and 2.36, δ(k n ) 1 = n 2 Z + f (K n) Z + (K n ) 1 = n 2 and thus equality holds. By Coollay 2.34, l + (K n ) 1. The set B = ({v 1, v 2,..., v n 2 }, {v n 1 }) is thus an optimal factional PSD focing set fo K n, so Z + f (K n) = n 2 and l + (K n ) = 1. In each of the next fou examples, optimality of the exhibited factional PSD focing sets is obtained by application of Coollaies 2.33 and Example Fo any n 2, the set B = (, {v 1 }) is an optimal factional PSD focing set fo P n, whee V (P n ) = {v 1, v 2,..., v n } in path ode, so Z + f (P n) = 0 and l + (P n ) = 1. Example Fo any n 3, the set B = ({v 1 }, {v 2 }) is an optimal factional PSD focing set fo C n, whee V (C n ) = {v 1, v 2,..., v n } in cycle ode, so Z + f (C n) = 1 and l + (C n ) = 1. Example Let n 4 and conside the wheel on n vetices, W n, which is obtained by adding a vetex w adjacent to evey vetex of C n 1. If B = (D, L) is any optimal factional PSD focing set fo C n 1, then B = (D {w}, L) is an optimal factional PSD focing set fo W n, so Z + f (W n) = 2 and l + (W n ) = 1. Example Let p q 1 and conside K p,q, the complete bipatite gaph on patite sets P and Q with P = p and Q = q. Let D be a set containing any (q 1) elements of Q and let L be a set containing any one element of P ; then B = (D, L) is an optimal factional PSD focing set fo K p,q, so Z + f (K p,q) = q 1 and l + (K p,q ) = 1. As a final example, we conside the factional PSD focing numbe of a tee. Example Suppose that T is a tee of ode at least 2. We have Z + (T ) = 1 (cf. [5, Example ]), so Coollay 2.36 implies that 0 Z + f (T ) Z+ (T ) 1 = 0 and hence equality holds. Coollay 2.34 implies that l + (T ) 1; if we let L be any leaf of T, then B = (, L) is an optimal factional PSD focing set, so Z + f (T ) = 0 and l+ (T ) = Thee-colo intepetation of skew zeo focing In this section, we intoduce a thee-colo intepetation of the skew zeo focing game and use this to show that the skew zeo focing numbe and factional (zeo) focing numbe of a gaph ae equal. Using the thee-colo intepetation, we deive new esults petaining to skew zeo focing numbe and the associated coloing pocess. 19