Matrix Partitions of Split Graphs

Transcription

1 Matrix Partitios of Split Graphs Tomás Feder, Pavol Hell, Ore Shklarsky Abstract arxiv: v2 [cs.dm] 20 Ju 2013 Matrix partitio problems geeralize a umber of atural graph partitio problems, ad have bee studied for several stadard graph classes. We prove that each matrix partitio problem has oly fiitely may miimal obstructios for split graphs. Previously such a result was oly kow for the class of cographs. (I particular, there are matrix partitio problems which have ifiitely may miimal chordal obstructios.) We provide (close) upper ad lower bouds o the maximum size of a miimal split obstructio. This shows for the first time that some matrices have expoetial-sized miimal obstructios of ay kid (ot ecessarily split graphs). We also discuss matrix partitios for bipartite ad co-bipartite graphs. 1 Itroductio The approach to graph partitio problems, proposed i [9, 2, 5], ad used i this paper, is iformed by the followig distictio betwee differet partitio problems. There are graph partitio problems which may be solved i polyomial time ad for which the set of miimal o-partitioable graphs is fiite. The split graphs recogitio problem is a well-kow example [8]. O the other had there are partitio problems, such as the bipartitio problem, which may be solved i polyomial time [10], but for which the set of miimal o-partitioable graphs is ifiite (i the case of the bipartitio problem, these are the odd cycles). Fially, there are umerous NP-complete graph partitio problems, such as the 3-colourig problem. Whe discussig classes of partitio problems, we will use patters to describe the requiremets of a partitio. I particular, the patters we examie specify partitio problems i which the iput graph s vertices are to be partitioed ito 268 Waverley St., Palo Alto, CA 94301, USA; tomas@theory.staford.edu School of Computig Sciece, Simo Fraser Uiversity, Buraby, B.C., Caada, V5A 1S6 School of Computig Sciece, Simo Fraser Uiversity, Buraby, B.C., Caada, V5A 1S6 1

2 1 INTRODUCTION idepedet sets, or cliques, or some combiatio of idepedet sets ad cliques. Further, we might require that two parts of vertices i the partitio be completely adjacet, or completely o-adjacet. Formally, we use matrices to describe these patters. Let M be a symmetric m m matrix over 0,1,. A M-partitio of a graph G is a partitio of the vertices of G ito parts P 1,P 2,...,P m such that two distict vertices i parts P i ad P j (possibly with i = j) are adjacet if M(i,j) = 1, ad oadjacet if M(i, j) = 0. The etry M(i, j) = sigifies o restrictio. Note that whe i = j these restrictios mea that part P i is either a clique, or a idepedet set, or is urestricted, whe M(i,i) is 1, or 0, or, respectively. Further, some of the parts may be empty. We may therefore assume that o of the diagoal etries of M are asterisks or else the problem is trivial. For a fixed matrix M, the M-partitio problem asks whether or ot a iput graph G admits a M-partitio. If a graph G fails to admit a M-partitio, we say that G is a M-obstructio. Further, if G is a M-obstructio but deletig ay vertex of G results i a M-partitioable graph, the G is a miimal M-obstructio. Give a graph G ad lists L(v) {1,...,m}, with v V(G), the list M- partitio problem asks whether G admits a M-partitio respectig the lists. That is, a M-partitio of G such that, for every v V(G), the vertex v is placed i a part P i oly if i P i. Note that diagoal asterisks do ot make the problem trivial whe lists are ivolved. I this paper, we will focus o the o-list versio, ad will explicitly refer to the list versio whe it is discussed. For ay matrix M i this paper, we assume that there are k zero etries ad l oe etries o M s diagoal. By row ad colum permutatios, we may further assume that M(0,0) = M(1,1) =... = M(k,k) = 0 ad M(k + 1,k + 1) =... = M(k + l,k + l) = 1. Let A be the submatrix of M o rows 1,...,k ad colums 1,...,k; let B be the submatrix of M o rows k+1,...,m ad colums k + 1,...,k; ad let C be the submatrix of M o rows 1,...,k ad colums k +1,...,m. Whe M has o diagoal asterisks, k +l = m, ad we say that M is i (A, B, C)-block form. Feder et al. have show that if there are asterisks i block A or block B of a matrix M, the there are ifiitely may miimal M-obstructios [7]. Thus, whe discussig geeral graphs, we must restrict our attetio to matrices i which the oly asterisk etries (if ay) are i the block C. Such matrices are called friedly. Of these, for ay m m matrix M cotaiig o asterisk etries at all (i.e. havig oly etries i {0,1}), it has bee show that the largest miimal M-obstructio is of size (k +1)(l+1) [3]. Eve whe restricted to chordal graphs, there are matrices for which there are 2

3 1 INTRODUCTION t+1... t t 1 2t Figure 1.1 A matrix with a family of ifiitely may miimal obstructios. ifiitely may chordal miimal obstructios [1, 6]. Oe of these matrices ad a ifiite family of chordal miimal obstructios to this matrix, appear frequetly i relatio to other classes of graphs i this paper, ad so are listed i Figure 1.1. The obstructio family i this figure is i fact a family of iterval graphs, so that the matrix has ifiitely may iterval miimal obstructios. Noetheless, for ay matrix M, the M-partitio problem restricted to iterval graphs ca be solved i polyomial time [11]. Note that the family i Figure 1.1 is ot a family of split graphs, as each member cotais 2K 2 as a iduced subgraph. For geeral matrices M, all kow upper bouds o the size of miimal obstructios to M-partitio are expoetial [3, 4, 9]; however, i oe of these cases has it bee show that expoetial-sized miimal obstructios to M-partitio actually exist. This paper is orgaized as follows: I Sectio 2, we show that for ay m m matrix M, a split miimal M-obstructio has at most O(m 2 2 m ) vertices. This implies that ay M-partitio problem (without lists) is solvable i polyomial time whe the iput is restricted to split-graphs. Sectio 3 exhibits, for a particular class of m m matrices, a split miimal obstructio of size Ω(2 m ), demostratig that the expoetial upper boud derived i Sectio 2 is early tight. As oted above, this meas that the class of split graph obstructios is the first class with fiite miimal obstructios kow to cotai expoetially large obstructios. I sectio 4, we discuss graphs that admit other types of partitios, such as bipartite graphs ad co-bipartite graphs. It is show that for these classes also there are oly fiitely may miimal obstructios for ay matrix M. These graph classes (icludig the class of split graphs) have a atural commo geeralizatio, amely graphs whose vertex set may be partitioed ito k idepedet sets ad l cliques, sometimes called (k, l)-graphs. Split graphs are (1, 1)-graphs, bipartite graphs are (2, 0)-graphs, ad co-bipartite graphs are (0, 2)-graphs. By cotrast we show that whe k+l 3, there is a matrix M with ifiitely may miimal (k,l)- graph obstructios. Whe k 2, there are ifiitely may miimal (k,l)-graph obstructios that are chordal. 3

4 2 MATRIX PARTITIONS OF SPLIT GRAPHS 2 Matrix Partitios of Split Graphs I this sectio we prove the followig theorem. Theorem 2.1. If M is a matrix with o diagoal asterisks, ad k l, the there are fiitely may split miimal M-obstructios. A set of vertices H V(G) is said to be homogeeous i G if the vertices of V(G) H ca be partitioed ito two sets, S 1 ad S 2 such that every vertex of S 1 is adjacet to every vertex of H, ad o vertex of S 2 is adjacet to a vertex of H. The proof of Theorem 2.1 relies o the existece of large homogeous sets i M-partitioable split graphs. Propositio 2.2. Let A be a k k matrix whose diagoal etries are all zero. Let G A be a split graph that admits a A-partitio. The every part P of a A-partitio of G A cotais a homogeeous set i G A of size at least P 1 2 k 1. Proof. Suppose the parts of the A-partitio of G A are P 1,...P k. Let C I be a partitio of V(G A ) ito a clique C ad idepedet set I. Note that for 1 i k, we have that P i C 1, sice each P i is a idepedet set. Now, the vertices i the set P 1 I are o-adjacet to all but at most k 1 vertices, oe i each P i C, for 2 i k (see Figure 2.1). Assume without loss of geerality that P i C = 1 ad let u i P i C, for 2 i k. As each u i is either adjacet to at least half of the vertices of P 1 I, or o-adjacet to at least half of the vertices of P 1 I, a homogeeous set of size at least P k 1 ca be foud i P 1. Sice this argumet may be repeated for ay other part i the partitio, we have the desired coclusio. Figure 2.1 Structure of a k-partite split graph Propositio 2.3. Let B be a l l matrix whose diagoal etries are all 1. Let G B be a split graph that admits a B-partitio. The every part P of a B-partitio of G B cotais a homogeeous set i G B of size at least P 1 2 l 1. Proof. The result follows from Propositio 2.2, sice G B admits a B-partitio if ad oly if G B admits a B-partitio, ad the complemet of a split graph is a split graph. 4

5 2 MATRIX PARTITIONS OF SPLIT GRAPHS We also require the followig observatio. Fact 2.4. Let M be a (A,B,C)-block matrix ad let G be a split graph. If C has a asterisk etry, the G admits a M-partitio. Proof. If C has a asterisk, the M cotais the matrix ( 0 1 ) as a pricipal submatrix. Thus G admits this partitio by defiitio of split graphs, sice every other part may be empty. Proof of Theorem 2.1. Let M be a m m matrix, with k diagoal 0s ad l diagoal 1s. Assume without loss of geerality that k l. We show that the umber of vertices i a split miimal M-obstructio is at most 2 k 1 (k +l)(2k+3)+1 O(k 2 2 k ) Suppose for cotradictio that G is a miimal M obstructio with at least 2 k 1 (k+l)(2k+3)+2 vertices. By Fact 2.4, we may assume that the submatrix C has o asterisks. Pick a arbitrary vertex v ad cosider a partitio of the graph G v o at least 2 k 1 (k + l)(2k + 3) + 1 vertices. As there are k + l parts i the partitio, by the pigeohole priciple there is a part, call it P, of size at least 2 k 1 (2k +3)+1. This part P is either a idepedet set or a clique, ad each of these cases will be cosidered separately below. Either way, by Propositios 2.2 ad 2.3, P cotais a homogeeous set i A or B (depedig o whether P is a idepedet set or a clique) of size at least P 1 2 k 1 2k +3. Sice C has o asterisks, this set is homogeeous i G. Thus G v has a homogeeous set of size at least 2k + 3, ad so G has a homogeeous set H of size at least k + 2, sice by the pigeohole priciple at least k+2 of the vertices of P agree o v. Now let w H, cosider a partitio of G w, ad recall that P is either a idepedet set or a clique. Case 1. If P is a idepedet set, the so is H; hece, there are at least k + 1 idepedet vertices i G w. As there are l k clique parts i the partitio of G w, ad o two idepedet vertices of H may be placed i the same clique part, at least oe vertex w H {w} must be placed i a idepedet part P. Sice w is ot adjacet to w ad both vertices belog to H, w ca be added to P, cotradictig the miimality of G. Case 2. If P is a clique the H w is a clique of size at least k+1, ad so i the partitio of G w, at least oe vertex of H w falls i a clique part P. As i Case 1, w ca be added to P, cotradictig miimality. Sice every matrix M has fiitely may split miimal obstructios, there is a obvious polyomial time algorithm for the M-partitio problem. However, a 5

6 3 A SPECIAL CLASS OF MATRICES more efficiet algorithm is described i what follows. A matrix M is crossed if each o-asterisk etry i its block C belogs to a row or colum i C of o-asterisk etries. It has bee show that if M is a crossed matrix, the the list M-partitio problem for chordal graphs ca be solved i polyomial time [1]. Sice split graphs are chordal, the same result applies for split graphs, ad we ca use this to solve the M-partitio problem for split graphs i polyomial time, bearig i mid that by Fact 2.4, we may assume that the block C has o asterisks ad so M is crossed. Theorem 2.5. If G is a split graph ad M is ay matrix, the the M-partitio problem for G ca be solved i time O( kl ). Whe dealig with the M-partitio problem with lists, it is show i [1] that there is a matrix M for which the list M-partitio problem is NP-complete, eve whe restricted to chordal graphs. I fact, the graphs costructed i that reductio are split graphs so that this list M-partitio problem remais NP-complete eve for split graphs. 3 A Special Class of Matrices As see i Sectio 2, for ay m m matrix M, there is a expoetial upper boud o the size of a largest split miimal M-obstructio. I this sectio we show a family of matrices for which this boud is early tight. For k,t N, with 1 t k 1, let M k,t be a k k matrix with diagoal etries all zero, t oes i row k, symmetrically, t oes i colum k ad asterisks everywhere else. By permutig the rows ad colums of M k,t we assume without loss of geerality that the oe etries of row k are i colums k t,...,k 1 ad symmetrically, that the oe etries of colum k are i rows k t,...,k 1. See Figure 3.1 for some examples. Theorem 3.1. There exist k,t N such that for the matrix M k,t, the size of the largest split miimal M-obstructio is Ω(2 k 1 ). Proof. We choose values of k ad t so that the matrix M k,t has a split miimal obstructio of size at least ( π k 1 ) k 1 +2k 1 2 Choose k = 2+1 ad t = for some N, so that the matrix M k,t has 2+1 parts. Place oes i row ad colums, + 1,...2 as well as i colums 2+1 ad rows,+1,...,2. Let P deote the part i row ad colum 2+1, 6

7 3 A SPECIAL CLASS OF MATRICES P 1 P 2 P M 3,1 P 1 P 2 P 3 P P 1 P 2 P 3 P 4 P M 5,3 M 4,1 Figure 3.1 Matrices M k,t for k {3,4,5} ad t {1,3} ad desigate the parts that have a oe to P as restricted parts, R 1,...,R ad the remaiig parts as urestricted parts, U 1,...,U. See Figure U 1 U 2 U S ( 2 )... R 1 R 2 R 2 Stable set ( 0 0 ) Clique P Stable set ( ) edge o-edge... b B a d(a) = 4 b 2 B b B d(b ) = 2 Figure 3.2 The matrix M 2+1, (left) ad a obstructio G (right) The miimal obstructio G, depicted i Figure 3.2, has a special vertex a, ad 2 vertices formig a clique B, that are all adjacet to a (so that B {a} is a clique of size 2 + 1). Further, G has aother 2 vertices formig a idepedet set B such that for each b B there is a b B that is ot adjacet to b but is adjacet to every other vertex of B {a}. Call b ad b mates. Fially, G has a idepedet set S of size ( 2 ) such that for every subset B of B of size, there is exactly oe vertex s S adjacet to exactly the vertices of B. Note that G is a split graph sice B {a} is a clique ad B S is a idepedet set, as see i Figure

8 3 A SPECIAL CLASS OF MATRICES S ( 2 )... B 2 b a d(a) = 4 b 2 B b B d(b ) = 2 Figure 3.3 A split partitio for G. To see that G is ideed a obstructio, suppose otherwise, ad ote that B {a} is a clique of size 2 + 1, so each of its vertices must be placed i a differet part. Sice each vertex of B has a mate i B that is adjacet to a ad all of the other vertices i B, all parts other tha the part cotaiig a have size at least two i ay M k,t -partitio of G. Thus oly the part cotaiig a may be a sigleto. Further P must be the oly sigleto part, otherwise all of the restricted parts must be sigletos, sice G cotais o iduced C 4. Therefore a P. Now whichever vertices of B are placed i the urestricted parts, as i Figure 3.4, there is a vertex s S adjacet to exactly these vertices, ad so must be placed ito oe of the restricted parts. But as s is ot adjacet to a, it caot be placed i a restricted part, ad s ca t be added to P; hece, G is ot M k,t -partitioable. U 1 U 2 U b 1,b 1 b 2,b... 2 b,b s S b 1 b2... b R 1 R 2 R b +1,b +1b +2,b b 2,b 2 P a a Figure 3.4 A attempt to partitio G. 8

9 4 GENERALIZED SPLIT GRAPHS To argue that G is a miimal obstructio, we show that removig a vertex from oe of S,B,B, or {a} allows a partitio for the resultig graph: (i) For s S partitio G s as follows: map a to P, place each b B, together with its mateb B, i some part, takig care that eighbours of the missig s are placed i urestricted parts. Now each remaiig vertex of S has a urestricted part to go to. (ii) We cosider b B together with its mate b B. For G b, place a i P, place b s mate b i a urestricted part P b, ad place all of S ad all of B i P b. This is possible sice B S is a idepedet set. place the remaiig 2 1 vertices of B i the remaiig 2 1 parts arbitrarily. To partitio G b, place b i P, map a together with all of the vertices of S i a urestricted part P a, ad place each other pair of mates v,v from B ad B ito a part, differet from P ad P a. (iii) Fially, G a ca be partitioed usig the restricted ad urestricted parts oly, ot placig aythig i P. Place each b ad its mate b ito a part. Each s S is oly forbidde from out of the 2 parts ad so ca be placed somewhere. Now G has 2k 1+ ( ) ( k 1 k 1 = ) vertices, ad usig Stirlig s approximatio, we 2 get 2 k 1 π k 1 2 = 22 π ( ) 2 ( 22 1 c ) ( = 2k 1 1 2c ), where 1 π π k 1 k 1 9 < c < Therefore G is of size expoetial i k. 4 Geeralized Split Graphs Recall that split graphs ca be viewed as a special case of (k,l)-graphs - those graphs whose vertices ca be partitioed ito k idepedet sets ad l cliques. (Thus split graphs are the (1,1)-graphs.) I this sectio, we focus o(k,l)-graphs other tha the(1,1)-graphs. We begi with (2, 0)- ad (0, 2)-graphs, ad the discuss other (k, l)-graphs. Recall that the (2,0)-graphs are the bipartite graphs, while the (0,2)-graphs are the co-bipartite graphs. As it turs out, there are fiitely may bipartite or co-bipartite miimal obstructios, for ay matrix M. Theorem 4.1. For ay m m matrix M, there are fiitely may bipartite miimal obstructios ad fiitely may co-bipartite miimal obstructios. 9

10 4 GENERALIZED SPLIT GRAPHS To prove Theorem 4.1 we use a approach similar i ature to that used Sectio 2. Startig with bipartite graphs, ote that we may assume that the matrix ( 0 0 ) is ot a pricipal submatrix of the matrix M, or else the problem would be trivial. Propositio 4.2. Let M be a (A, B, C)-block matrix, with A of size k k ad B of size l l. Suppose the block A has o asterisk etries. If G is a M- partitioable bipartite graph, the ay part P of A i a M-partitio of G cotais a homogeeous set of size at least P 2 2l Proof. Fix a bipartitio of G ad let P be a part of A i a M-partitio of G. We argue that P has the desired size. As A has o asterisks, the vertices of P all have the same adjacecy relatio to vertices i other parts of A. Now let P be some part of B. Sice G is bipartite, P ca have at most two vertices, oe from each part of the bipartitio of G. Let these vertices be x ad y. By the pigeohole priciple, x is either adjacet to, or o-adjacet to, at least half of the vertices of P. Suppose with out loss of geerality, that x is adjacet to at least half of the vertices of P. Call these vertices P x. Applyig the pigeohole priciple agai, this time to the vertex y, we have that y is either adjacet to, or o-adjacet to, at least half of the vertices of P x. Let the larger of these two sets be P xy, ad ote that P xy P 2 2 Now there are l 1 clique parts other tha P, each of size at most two. Iductively, we obtai a homogeeous set i P of size at least P 2 2l Theorem 4.1 ow follows for bipartite graphs. The proof for co-bipartite graphs follows by complemetatio. Proof of Theorem 4.1. As discussed above, we assume that A cotais o asterisk etries. We show that ay bipartite miimal obstructio is of size at most 2 2l (k +l)(2l+3) Suppose otherwise, ad let G be a miimal obstructio with at least 2 2l (k + l)(2l+3)+1 vertices. For a arbitrary vertex v, the graph G v is M-partitioable, ad so some part P i a M-partitio of G v cotais at least 2 2l (2l+3) vertices. Sice 2 2l (2l+3) 3 for l 0, ad o clique part of M may cotai more tha two vertices, P must be a idepedet set. Thus by Propositio 4.2, P cotais a homogeeous set of size at least P 2 2l 2l + 3. By the pigeohole priciple, G has a homogeeous set H of size at least l+2. Note that H is a idepedet set. Let h H, ad cosider a partitio of G h. As there are oly l cliques ad l+1 vertices i H h, there must be a part P of A that cotais a vertex h of H h. But sice H is a idepedet set, ad h has the same eighbourhood as h, we may add h to P to obtai a partitio of G, a cotradictio. 10

11 REFERENCES We ow cosider (k,l)-graphs for values of k ad l that satisfy k + l 3. For coveiece, let (k,l) deote the set of (k,l)-graphs. The family of graphs depicted i Figure 1.1 is a ifiite family of chordal miimal obstructios to the matrix M 3,1 [6]. We defie the family more precisely as follows. For t 3, let G(t) be the graph cosistig of a eve path o 2t vertices, ad a additioal vertex u. u is adjacet to all vertices of the path, except the edpoits. Note that each G(t) is chordal. Theorem 4.3. If k,l N such that k+l 3, the there exists a matrix M that has ifiitely may (k,l)-miimal obstructios. Proof. Note that for ay t 3, G(t) is 3-colourable, ad G(t) is partitioable ito a bipartite graph ad a clique. That is, G(t) (3,0) (2,1). Therefore, for the matrixm 3,1, there are ifiitely may (chordal) miimal(2,1) (3,0) obstructios. By complemetatio, for ay t 3, the graph G(t) is i (1,2) (0,3), providig ifiitely may (chordal) (1,2) (0,3) obstructios for the matrix M 3,1. Now if k 1, the sice k + l 3, it must be that l 2, ad so the family {G(t) t 3} is a family of (k,l)-miimal obstructios for M 3,1. O the other had, if k 2, the the family {G(t) t 3} is a family of (k,l)-miimal (chordal) obstructios for the matrix M 3,1. Refereces [1] T. Feder,, P. Hell, S. Klei, L. T. Nogueira, ad F. Protti. List Matrix Partitios of Chordal Graphs. Theoretical Computer Sciece, 349:52 66, [2] T. Feder ad P. Hell. Matrix Partitios of Perfect Graphs. Discrete Mathematics, 306(19-20): , October [3] T. Feder ad P. Hell. O Realizatios of Poit Determiig Graphs, ad Obstructios to Full Homomorphisms. Discrete Mathematics, 308(9): , [4] T. Feder, P Hell, ad W. Hochstättler. Geeralized Colourigs (Matrix Partitios) of Cographs. I Graph Theory i Paris, pages Birkhauser Verlag, [5] T. Feder, P. Hell, Sulamita Klei, ad Rajeev Motwai. List Partitios. SIAM Joural o Discrete Mathematics, 16(3):449, November [6] T. Feder, P. Hell, ad S. Rizi. Obstructios to Partitios of Chordal Graphs. accepted i Discrete Mathematics. 11

12 REFERENCES [7] T. Feder, P. Hell, ad W. Xie. Matrix Partitios with Fiitely May Obstructios. The Electroic Joural of Combiatorics, 14, [8] S. Földes ad P.L. Hammer. Split Graphs. Proc. 8th Southeaster Cof. o Combiatorics, Graph Theory ad Computig (F. Hoffma et al., eds.), Louisiaa State Uiv., Bato Rouge, Louisiaa. (As cited i Golumbic 2004), pages , [9] P. Hell. Graph Partitios with Prescribed Patters. accepted i Europea Joural of Combiatorics. [10] D. Köig. Theorie der Edliche ud Uedliche Graphe (as cited i West, 2001) [11] P. Valadkha. Graph Partitios. PhD thesis, Simo Fraser Uiversity,