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1 Slightly Triangulated Graphs Are Perfect Frederic Maire frm@ccr.jussieu.fr Case 189 Equipe Combinatoire Universite Paris 6, France December 21, 1995 Abstract A graph is triangulated if it has no chordless cycle with at least four vertices (8k 4; C k 6 G). These graphs have been generalized by R. Hayward with the weakly triangulated graphs (8k 5; C k ; Ck 6 G). In this note we propose a new generalization of triangulated graphs. A graph G is slightly triangulated if it satises the two following conditions : 1. G contains no chordless cycle with at least 5 vertices. 2. For every induced subgraph H of G, there is a vertex in H the neighbourhood of which in H contains no chordless path of 4 vertices. We shall prove that these graphs are perfect, and compare them with other classical families of perfect graphs. 1 Introduction Let P k be the chordless path with k vertices and C k be the chordless cycle with k vertices. Let G denote the complement of the graph G. We write H G if H is an induced subgraph of G. For a graph G the maximum size of a clique is denoted by!(g), the maximum size of a stable set (G), the number of vertices n. Claude Berge [1] denes a graph G to be perfect 1

2 if for every induced subgraph H of G the chromatic number of H is equal to the largest size of a clique in H. The triangulated graphs constitute a wide class of perfect graphs with numerous applications. Moreover ecient algorithms are known for these graphs. Finding a maximum stable set, a maximum clique, a minimal clique cover, a minimal colouring and recognizing triangulated graphs can be done in polynomial time. The reader is referred to [7] for further information. Every triangulated graph G satises the two following properties : 1. 8k 4; C k 6 G 2. 8H G; 9x 2 H;? H (x) 6 P2 (i.e,? H (x) is a complete subgraph). In fact the second property is a consequence of the rst one. A vertex x with a P2 -free neighbourhood is called simplicial. The denition of a slightly triangulated graph G can be seen as a relaxation of these conditions. 1. 8k 5; C k 6 G 2. 8H G; 9x 2 H;? H (x) 6 P4 = P 4 A vertex whose neighbourhood is P 4 -free is called a P 4 -free vertex. Weakly triangulated and slightly triangulated graphs are not comparable as we shall see below. The reader interested in weakly triangulated graphs is referred to [8] and [9] for further information. 2 Main result The following result due to Seinsche [15] will be used several times. Lemma 1 If a graph H is P 4 -free then H or H is disconnected. Another key tool is the star cutset : this is a cutset C such that some vertex in C is adjacent to all the remaining vertices in C. V. Chvatal has shown in [6] that a minimal imperfect graph cannot have a star cutset. Lemma 2 If G is a minimal imperfect graph then neither G nor its complement G has a star cutset. 2

3 The proof for the perfectness of a slightly triangulated graph relies upon some properties of partitionable graphs. See [2] for an introduction to this subject. The proofs of the results used here on partitionable graphs can be found in [4] or [5]. We rst recall some properties of partitionable graphs. A graph G is (;!)-partitionable if there exist integers 2;! 2, such that jv (G)j =! + 1 and for every v 2 V (G) there exists a partition of V (G) n v into stable sets of size and another partition into cliques of size!. An immediate consequence of Lovasz' characterization of perfect graphs is that every minimal imperfect graph G is partitionable with = (G) and! =!(G). Padberg [14] has shown that if G is minimal imperfect, then G has exactly n cliques of size!, every vertex of G is in exactly! cliques of size!, and the n n incidence matrix of!-cliques versus vertices of G is nonsingular. Moreover the maximum cliques and the maximum stables can be numbered C 1 ; C 2 ; : : : ; C n and S 1 ; S 2 ; : : : ; S n in such a way that C i \ S j = ; i i = j Note that the complement of a partitionable graph is partitionable. We have the following generalization of the starcutset lemma. Lemma 3 If G is a partitionable graph then G has no star cutset. Proof: Suppose G has a starcutset T with center x. Let A 1 ; A 2 ; : : : ; A k be the components of G n T. As every proper subgraph of G is!-colorable. The subgraphs G n A 1 and T [ A 1 are!-colorable. Let B 1 be the color class of x in G n A 1, and B 2 the color class of x in T [ A 1. Then B 1 [ B 2 is a stable set, which meets (has a non-empty intersection with) every maximum clique of G. Let y be a vertex of G n (B 1 [ B 2 ). As there is a partition of G n y in!-cliques, we have jb 1 [ B 2 j =. So we have found a maximum stable, which meets every maximum clique. A contradiction. 2 Lemma 4 A graph (;!)-partitionable G has no P 4 -free vertex v such that all the (!? 1)-cliques of? G (v) belong to the same component of? G (v). Proof: We assume that G has a P 4 -free vertex v and show that this leads to a contradiction. Let H be the neighbourhood of v. Case 1 If H is disconnected then v [? G(v) is a star cutset of G. A contradiction. 3

4 Case 2 If H is connected then (by Lemma 1) H is disconnected. If all maximum cliques are in the same component A of H, then remove from G all the edges between v and the other components of H and call this new graph G 0. According to [5], if we delete edges which belong to no!-clique of a (;!)-partitionable we obtain another (;!)-partitionable graph. The graph G 0 is therefore (;!)-partitionable. We apply the same argument as in Case 1, to obtain a contradiction: The neighbourhood of v in G 0 is A, which is a P 4 -free and connected. Therefore A is disconnected. Thus, we have v [? G 0(v) is a star cutset of G0. A contradiction. 2. Theorem 1 Slightly triangulated graphs are perfect Proof: Let G be a slightly triangulated and minimal imperfect graph, we shall show that this assumption leads to contradictions. Let v be a vertex such that P 4 6? G (v). Remember that if a graph H is P 4 -free then either H or H is disconnected. If? G (v) is disconnected then v [? G(v) is a star cutset of G. Thus we can assume that?g (v) is disconnected. Let? G (v) = N 1 [ N 2 be a bipartition such that 8n 1 ; n 2 2 N 1 N 2 ; n 1 n 2 62 E. Because of Lemma 4 we can assume that there are (!?1)-cliques in both N 1 and N 2. Let I denote the set of vertices of G n (v [? G (v)) adjacent to N 1 and N 2. i.e. I = ((? G (N 1 ) n N 1 ) \ (? G (N 2 ) n N 2 )) n v Claim 1 8x 2 I; N 1 6? G (x); N 2 6? G (x) Proof: Suppose for example that N 2?(x). Then N 1 6?(x). Otherwise x and v would have comparable neighbourhoods (A minimal imperfect cannot have such a pair of vertices, otherwise, the incidence matrix of!-cliques versus vertices of G would have identical rows for x and v, and therefore would be singular. But we know that these incidence matrices are regular). As v [ (?(x) \?(v)) is not a star cutset, there is a chordless chain (included in G n (v [ (?(x) \?(v)))), whose length is 2 joining x to N 1 n?(x). Let y 1 ; y 2 ; : : : ; y k = x be this chain. We have k 3, y 1 2 N 1, and 8i 2; y i 62?(v). We will consider two cases. We abbreviate? G (z) \ N i by N i (z). case 1 N 2 n N 2 (y 2 ) 6= ;. Let z 2 N 2 n N 2 (y 2 ). Let l be the smallest index such that y l z 2 E, then fv; y 1 ; y 2 ; : : : ; y l ; zg induces a chordless cycle with a length 5. case 2 N 2 (y 2 ) = N 2. Then k 4, otherwise there would be a (! + 1)-clique 4

5 in x [ y 2 [ N 2. We can choose x such that j?(x) \?(v)j is maximum among the vertices of I. Therefore 9t 2 N 1 (x) n N 1 (y 2 ), since N 1 (y 2 ) n N 1 (x) 6= ;. If ty 1 2 E then fy 1 ; t; x; n 2 ; y 2 g = C 5, where n 2 is any vertex of N 2. If ty 1 62 E then let l be the smallest index such that y l t 2 E, then l 3 and fv; y 1 ; y 2 ; : : : ; y l ; tg = C l+25. This ends the proof of the claim. Claim 2 The intersection I is nonempty. Proof: Suppose that I = ;, since v is not a cutset there exists a chordless path from N 1 to N 2. But this would imply that G contains a chordless cycle with 5 or more vertices. A contradiction. Consider x 2 I such that j?(x) \?(v)j is maximum. According to the last claim N 1 6?(x) and N 2 6?(x). Let x 1 ; x 2 ; : : : ; x k = x be a chordless chain joining x to N 1 n?(x) in G n (v [ N 1 (x) [ N 2 ). The existence of this chain is guaranted by the star cutset lemma. We have k 3, x 1 2 N 1, and 8i 2; x i 62?(v). We have N 2 (x 2 ) N 2 (x), otherwise let z 2 N 2 (x)nn 2 (x 2 ), and let l be the smallest index such that x l z 2 E, then fv; x 1 ; x 2 ; : : : ; x l ; zg = C l+25. As j?(x) \?(v)j is maximum, 9u 2 N 1 (x) nn 1 (x 2 ). We have x 1 u 2 E otherwise 9l; fv; x 1 ; x 2 ; : : : ; x l ; ug = C l+25. If k 4 then xx 2 62 E and fx 1 ; u; x; n 2 ; x 2 g = C 5, where n 2 2 N 2 (x). To sum up, there is a chordless chain joining x to N 1 n N 1 (x) whose length equals 2 and with the properties found above on the inclusions of the neighbourhoods. The same holds for N 2 n N 2 (x). Let fx; b 1 ; a 1 g be a chordless chain from x to N 1 nn 1 (x), and let fx; b 2 ; a 2 g be a chordless chain from x to N 2 n N 2 (x). As N 2 (b 1 ) N 2 (x) and N 2 (x) n N 2 (b 2 ) 6= ;, we must have b 1 6= b 2. Let u 1 2 N 1 (x) n N 1 (b 1 ), and Let u 2 2 N 2 (x)nn 2 (b 2 ). The vertices u i and a i are adjacent, otherwise fv; a i ; b i ; x; u i g = C 5. To end the proof of the theorem we have two cases to consider. case 1 a 1 b 2 ; a 2 b 1 62 E. case 1.1 b 1 b 2 2 E. then fv; a 1 ; b 1 ; b 2 ; a 2 g = C 5. case 1.2 b 1 b 2 62 E. then fv; a 1 ; b 1 ; x; b 2 ; a 2 g = C 6. case 2 a 1 b 2 2 E for example. then fv; a 1 ; b 2 ; x; u 2 g = C

6 3 Comparison with other classes of perfect graphs Berge and Duchet [3] call a graph G strongly perfect if every induced subgraph has a stable set which meets every maximal cliques. An even-pair is a pair of (non-adjacent) vertices such that every chordless path which joins the two vertices has an even number of edges. A graph G is said to be strict quasiparity if every induced subgraph H of G is a clique or has an even-pair. A graph G is said to be quasi-parity if for every induced subgraph H of G there is an even-pair in H or H. A graph is called Meyniel [13] if every odd cycle with ve or more vertices has two chords. A graph is Raspail if every odd cycle of length at least 5 has a chord joining two vertices at distance 2 on the cycle. The perfection of Raspail graphs is still an open problem. Nevertheless, Anna Lubiw [10] proved that Raspail graphs such that every induced subgraph contains a P 4 -free vertex are perfect. The graph obtained from C 9 by adding the edges (2; 5), (5; 8) and (8; 2) is sligthly triangulated but not Raspail. Meyniel graphs are strongly perfect and strict quasi-parity. Weakly triangulated graphs are strict quasi-parity. The fact that C6 is neither strongly perfect, nor strict quasi-parity implies that slightly triangulated are not necessarily weakly triangulated or Meyniel. Let A and B be two P 4. The graph obtained by joining every vertex of A to every vertex of B is weakly triangulated, but not slightly triangulated. The fact that C 6 is Meyniel implies that strongly perfect and strict quasiparity graphs are not necessarily slightly triangulated. The graph C 6 belongs also to the class dened by Lubiw. We do not know if slightly triangulated graphs are quasi-parity. 4 Open problems In [11] we presented polynomial algorithms for recognizing slightly triangulated graphs and for nding a maximum clique. We proposed also an algorithm for nding an optimal colouring. Unfortunately this colouring algorithm is not polynomial. As slightly triangulated graphs appear naturally in the study of the intersection graphs of the maximal rectangles of orthogonal polygons [12] it would be interesting to nd an ecient way of colouring 6

7 slightly triangulated graphs. A hole is a chordless cycle with 4 or more vertices. An antihole is the complement of a hole. Berge's strong conjecture (G is perfect i G and G have no odd hole) implies that a minimal imperfect graph with no odd hole is an odd antihole with at least 7 vertices. We conjecture the following : 'A minimal imperfect graph with no odd hole has no P 4 -free vertex'. Acknowledgements The author thanks Frederic Maray for simplifying the proof of Lemma 4 and for his suggestions. The author is also grateful to Myriam Preissman for reading an early version of this paper. References [1] Berge, C. : Sur une conjecture relative au probleme des codes optimaux, Commun. 13 eme Assemblee Gen. URSI, Tokyo, [2] Berge, C. Chvatal, V.:Topics on perfect graphs, Ann. Discrete Math. 21, (1984) [3] C. Berge and P. Duchet, strongly perfect graphs, in "Topics on perfect graphs" (C. Berge and V. Chvatal, Eds.), pp 57-61, North-Holland, Amsterdam, [4] R.G. Bland, H.C. Huang and L.E. Trotter, Jr. Graphical properties related to minimal imperfection, in "Topics on perfect graphs" (C. Berge and V. Chvatal, Eds.), pp , North-Holland, Amsterdam, [5] V. Chvatal, R.L. Graham, A.F Perold and S.H. Whitesides, Combinatorial designs related to the perfect graph conjecture, in "Topics on perfect graphs" (C. Berge and V. Chvatal, Eds.), pp , North-Holland, Amsterdam, [6] V. Chvatal, Perfectly Ordered Graphs, in "Topics on perfect graphs" (C. Berge and V. Chvatal, Eds.), pp 63-65, North-Holland, Amsterdam,

8 [7] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, [8] Hayward, R.:Weakly triangulated graphs. J. Comb. Theory(B) 39, (1985) [9] Hayward, R. Hoang, C. Maray, F.:Optimizing weakly triangulated graphs, Graphs and Combinatorics 5, (1989) [10] A. Lubiw:Short-chorded and perfect graphs, J. Comb. Theory(B) 39, 24-33(1991) [11] Maire F. : Des polyominos aux graphes parfaits, PhD Thesis University Paris 6, june [12] Maire F. : A characterization of intersection graphs of the maximal rectangles of a polyomino, Discrete Maths 120 (1993), 211{214 [13] H. Meyniel, The graphs whose odd cycles have at leat two chords, in "Topics on perfect graphs" (C. Berge and V. Chvatal, Eds.), pp , North-Holland, Amsterdam, [14] M. Padberg (1974) "Perfect zero-one matrices", Math. Programming, 6, [15] D. Seinsche, On a property of the class of n-colorable graphs, J. Comb. Theory(B) 16, (1974). 8

Small Survey on Perfect Graphs

Small Survey on Perfect Graphs Michele Alberti ENS Lyon December 8, 2010 Abstract This is a small survey on the exciting world of Perfect Graphs. We will see when a graph is perfect and which are families