- Email: [email protected]
Brambles and independent packings in chordal graphs
Discrete Mathematics 309 (2009) 5766–5769
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Brambles and independent packings in chordal graphsI Kathie Cameron Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
article
info
Article history: Received 31 August 2006 Accepted 28 May 2008 Available online 9 July 2008 Keywords: Independent packing Induced matching Bramble Chordal graph Min–max theorem Independent set Clique
abstract An independent packing of triangles is a set of pairwise disjoint triangles, no two of which are joined by an edge. A triangle bramble is a set of triangles, every pair of which intersect or are joined by an edge. More generally, I consider independent packings and brambles of any specified connected graphs, not just triangles. I give a min–max theorem for the maximum number of graphs in an independent packing of any family of connected graphs in a chordal graph, and a dual min–max theorem for the maximum number of graphs in a bramble in a chordal graph. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Throughout this paper, H is a fixed set of connected graphs. An H -subgraph of a given graph G is a subgraph of G isomorphic to a member of H . An H -packing of a given graph G is a pairwise node-disjoint set of H -subgraphs of G. An independent H -packing of a given graph G is an H -packing of G in which no two subgraphs of the packing are joined by an edge of G. When H is a single graph H, we have H-packings and independent H-packings of G. Thus a K2 -packing of G is precisely a matching of G, and an independent K2 -packing of G is precisely an induced matching of G. Similarly, an independent K1 -packing of G is precisely an independent set of nodes in G. The size of an independent packing is the number of graphs in the packing. The independent H -packing problem asks for the maximum number of graphs in an independent H -packing; that is, the maximum size of an independent H -packing. The problem of finding a largest induced matching, i.e., the independent K2 -packing problem, has been widely studied [6, 7,9–14,16,17,23,25,27]. It is known to be NP-hard for bipartite graphs [6,29], and for planar graphs [24]; polynomial-time algorithms for various classes of structured graphs were given in [6,7,9,11,16,17]. In [8], we showed that the independent packing problem is polynomial-time solvable for many classes of structured graphs, including weakly chordal graphs, asteroidal triple-free graphs, polygon-circle graphs, and interval-filament graphs. These classes contain other well-known classes such as chordal graphs, cocomparability graphs, circle graphs, circular-arc graphs, and outer-planar graphs. (See [4] for definitions of these classes of graphs.) Recall that a graph is called chordal if it is the intersection graph of a set of subtrees of a tree. In this paper, I give a min–max theorem for the maximum number of graphs contained in an independent H -packing of a given chordal graph G. This extends the following min–max theorem given in [6] for the maximum size of an induced matching in a chordal graph. To state the theorem, we need the following definition. A K2 -sunflower is a clique together with some incident edges. (In [6], I called these clique-neighbourhoods.) Please note that in this paper, cliques are not necessarily maximal.
I Research supported by the Natural Sciences and Engineering Research Council of Canada.
E-mail address: [email protected]. 0012-365X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2008.05.031
K. Cameron / Discrete Mathematics 309 (2009) 5766–5769
5767
Theorem 1 ([6]). For any chordal graph G, the maximum size of an induced matching in G equals the minimum number of K2 -sunflowers which together cover all the edges of G. More generally, I define an H -sunflower of a graph G to be a clique C of G together with a set P of H -subgraphs of G each of which intersects C ; C is called the centre of the sunflower and P is called the set of petals of the sunflower. The size of an H -sunflower is the number of H -subgraphs it contains. We say an H -sunflower, S, of G covers a subgraph H of G if H is a subgraph of S. The main result of this paper is the following. Theorem 2. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of H -sunflowers which together cover all the H -subgraphs of G. Theorem 2 can also be stated in the following way. Theorem 3. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of cliques of G which meet all the H -subgraphs of G. To see that Theorems 2 and 3 are equivalent, note that if S is a set of H -sunflowers of graph G which together cover all the H -subgraphs of G, then the set of centres of S is a set of cliques which meet all the H -subgraphs of G. Conversely, given a set C of cliques of G which meet all the H -subgraphs of G, if for each clique C ∈ C , we consider the sunflower S (C ) consisting of C together with all H -subgraphs of G which meet C, then {S (C ) : C ∈ C } is a set of H -sunflowers which together cover all the H -subgraphs of G. Hell, Klein, Tito-Nogueira and Protti [19–22] proved Theorem 3 in the case that H is a single fixed complete graph Kr , r ≥ 3. Given a graph G and set H of graphs, we define another graph, H (G), whose nodes are the H -subgraphs of G, and such that two nodes are adjacent exactly when the subgraphs they correspond to intersect or are joined by an edge of G. Then independent H -packings in G correspond precisely to independent sets of nodes in H (G). Where H = {K2 }, H (G) is the square of the line-graph of G. In [8], we proved the following theorem. Theorem 4 ([8]). If G is a chordal graph and H is a fixed set of connected graphs, then H (G) is a chordal graph. Chordal graphs are perfect, and from the perfect min–max theorems for chordal graphs due to Hajnal and Surányi [18] and Berge [2] together with Theorem 4, we obtain the following min–max theorems. An independent H -packing I covers an H -subgraph H if H ∈ I. Theorem 5. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of cliques of H (G) which together cover all the nodes of H (G). Theorem 6. For any chordal graph G and fixed set H of connected graphs, the maximum size of a clique in H (G) equals the minimum number of independent H -packings of G which together cover all the H -subgraphs of G. Theorems 5 and 6 will be more clear if we can characterize cliques in H (G). Reed [28] defines a bramble in graph G to be a collection of node subsets of G, each of which induces a connected subgraph of G, and every pair of which either intersect or are joined by an edge of G. Brambles are important in the study of treewidth [1,3,28]. For a fixed set H of connected graphs, I define an H -bramble in a graph G to be a set S of H -subgraphs of G, every pair of which either intersect or are joined by an edge of G. The size of the bramble is |S |. Clearly, the set of petals of any H -sunflower is an H -bramble. More generally, the set of all H -subgraphs of an H sunflower is an H -bramble. It is easy to see that H -brambles in G correspond precisely to the node-sets of cliques in H (G). An H -bramble B covers an H -subgraph H if H ∈ B . Thus Theorems 5 and 6 become the following. Theorem 7. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of H -brambles of G which together cover all the H -subgraphs of G. Theorem 8. For any chordal graph G and fixed set H of connected graphs, the maximum size of an H -bramble in G equals the minimum number of independent H -packings of G which together cover all the H -subgraphs of G. In order to better understand Theorems 7 and 8, I now give a characterization of H -brambles in chordal graphs. Theorem 9. Let G be a chordal graph and H a fixed set of connected graphs. Every H -bramble is contained in some H -sunflower.
5768
K. Cameron / Discrete Mathematics 309 (2009) 5766–5769
Proof. Consider a representation of G as the intersection graph of a collection F of subtrees of a tree T . Tree T can be chosen so that the nodes of T correspond to the maximal cliques in G, and so that where Tv is the subtree of T which represents v ∈ V (G), the nodes of Tv correspond to the maximal cliques of T which contain v [5,15,30]. Let N be an H -bramble, say N = {Si : 1 ≤ i ≤ k}, where each Si is an H -subgraph of G. Since subgraph Si is connected, Ti = ∪{Tv : v ∈ V (Si )} is also a subtree of T . Let F 0 = {Ti : 1 ≤ i ≤ k}. Because N is a bramble, for every i and j, 1≤ i < j ≤ k, Ti ∩ Tj 6= Φ . Then by the Helly property which holds for subtrees of a tree, ∩F 0 6= Φ . Say c ∈ ∩F 0 , where c ∈ V (T ). Then c corresponds to a maximal clique C in G, and for every i, since c ∈ ∩F 0 = ∩{Ti : 1 ≤ i ≤ k}, it follows that c ∈ Ti , and thus c ∈ Tv for some v ∈ V (Si ), and thus v ∈ C ; so C intersects each Si . Thus the graph formed by ∪N is contained in the H -sunflower consisting of C together with {Si : 1 ≤ i ≤ k}, which is contained in the H -sunflower consisting of C together with the H -subgraphs of G which intersect C . Corollary 1. In a chordal graph, every maximal H -bramble corresponds to a maximal H -sunflower, and conversely. Theorem 7 together with Corollary 1 gives Theorem 2. Theorem 8 together with Corollary 1 gives the following theorem. Theorem 10. For any chordal graph G and fixed set H of connected graphs, the maximum size of an H -sunflower equals the minimum number of independent H -packings which together cover all the H -subgraphs of G. Where H = {K2 }, Theorem 10 was proved in [6]. Given a graph G and a weighting w = (wv : v ∈ V (G)) of V (G), the weight of an independent set of nodes is the sum of the weights of the nodes in the independent set. The weight of a clique is the sum of the weights of the nodes in the clique. Given a graph G, a fixed set H of connected graphs, and a weighting w = (wH : H ∈ H ) of H , the weight of an independent H -packing is the sum of the weights of the graphs in the packing, and the weight of an H -sunflower is the sum of the weights of its H -subgraphs. Weighted versions of the perfect graph min–max theorems hold [26], and are stated below as Theorems 11 and 12. These give Theorems 13 and 14 below, which are weighted versions of Theorems 2 and 10. Theorem 11 ([26]). Let K be a perfect graph with node-set V (K ). For any integer-valued weighting w = (wv : v ∈ V (K )) of V (K ), max weight of an independent set in K
)
( X
= min
X
yC : ∀ nodes v of V (K ),
yC ≥ wv ; ∀ cliques C of K , yC ≥ 0, yC integer
.
v is in clique C
cliques C of K
Theorem 12 ([26]). Let K be a perfect graph with node-set V (K ). For any integer-valued weighting w = (wv : v ∈ V (K )) of V (K ), max weight of a clique in K
(
) X
= min
X
yI : ∀ nodes v of V (K ),
indep. sets I of K
yI ≥ wv ; ∀ indep. sets I of K , yI ≥ 0, yI integer
.
v is in indep. set I
Apply Theorem 11 to the graph K = H (G), where G is a chordal graph and where H is a fixed set of connected graphs. Then K = H (G) is a chordal graph, and thus is perfect. Also, H -subgraphs of G correspond to nodes of K = H (G); independent H -packings in G correspond to independent sets of nodes in K = H (G); and H -sunflowers of G correspond to cliques of K = H (G). Thus Theorem 11 gives the following theorem. Theorem 13. Let G be a chordal graph with node-set V (G). Let H be a fixed set of connected graphs. For any integer-valued weighting w = (wH : H ∈ H ) of H , max weight of an independent H -packing in G
= min
X
H -sunflowers S
yS : ∀H -subgraphs H of G,
X
yS ≥ wH ; ∀H -sunflowers S of G, yS ≥ 0, yS integer
H is a subgraph of sunflow er S
.
Similarly, applying Theorem 12 to the graph K = H (G), where G is a chordal graph gives the following theorem. Theorem 14. Let G be a chordal graph with node-set V (G). Let H be a fixed set of connected graphs. For any integer-valued weighting w = (wH : H ∈ H ) of H ,
K. Cameron / Discrete Mathematics 309 (2009) 5766–5769
5769
max weight of an H -sunflower in G
= min
X
indep. H -packings I
yI : ∀H -subgraphs H of G,
X H is in indep. packing I
yI ≥ wH ; ∀ indep.H -packings I of G, yI ≥ 0, yI integer
.
Where H = {K2 }, Theorems 13 and 14 were proved in [6]. References [1] Patrick Bellenbaum, Reinhard Diestel, Two short proofs concerning tree decompositions, Combinatorics, Probability and Computing 11 (2002) 541–547. [2] Claude Berge, Les problèmes de colorations en théorie des graphs, Publications Institut Statist. University Paris 9 (1960) 123–160. [3] Hans Bodlaender, Alexander Grigoriev, Arie Koster, Treewidth lower bounds with brambles, Algorithmica 51 (2008) 81–98. [4] Andreas Brandstädt, Van Bang Le, Jeremy P. Spinrad, Graph Classes - A Survey, SIAM Publications, Philadelphia, PA, 1999. [5] P. Buneman, A characterization of rigid circuit graphs, Discrete Mathematics 9 (1974) 205–212. [6] Kathie Cameron, Induced matchings, Discrete Applied Mathematics 24 (1989) 97–102. [7] Kathie Cameron, Induced matchings in intersection graphs, Discrete Mathematics 278 (2003) 1–9. [8] Kathie Cameron, Pavol Hell, Independent packings in structured graphs, Mathematical Programming Series B 105 (2006) 201–213. [9] K. Cameron, R. Sritharan, Y. Tang, Finding a maximum induced matching in weakly chordal graphs, Discrete Mathematics 266 (2003) 133–142. [10] K. Cameron, T.L. Walker, The graphs with maximum matching and maximum induced matching the same size, Discrete Mathematics 299 (2005) 49–55. [11] Jou-Ming Chang, Induced matchings in asteroidal-triple free graphs, Discrete Applied Mathematics 132 (2003) 67–78. [12] W. Duckworth, N.C. Wormald, M. Zito, Maximum induced matchings of random cubic graphs, Journal of Computational and Applied Mathematics 142 (2002) 39–50. [13] R.J. Faudree, A. Gyárfás, R.H. Schelp, Zs. Tuza, Induced matchings in bipartite graphs, Discrete Mathematics 78 (1989) 83–87. [14] G. Fricke, R.C. Laskar, Strong matchings on trees, Congressus Numerantium 89 (1992) 239–243. [15] F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, SIAM Journal on Computing 1 (1972) 47–56. [16] M.C. Golumbic, R.C. Laskar, Irredundancy in circular-arc graphs, Discrete Applied Mathematics 44 (1993) 79–89. [17] M.C. Golumbic, M. Lewenstein, New results on induced matchings, Discrete Applied Mathematics 101 (2000) 157–165. [18] A. Hajnal, J. Surányi, Über die Auflösung von Graphen in vollständige Teilgraphen, Ann. University Sci. Budapest, Eötvös, Sect. Math 1 (1958) 113–121. [19] P. Hell, S. Klein, F. Protti, L. Tito, On generalized split graphs, Electronic Notes in Discrete Mathematics 7 (2001) www.elsevier.com. [20] P. Hell, S. Klein, L.T. Nogueira, F. Protti, Independent Kr ’s in chordal graphs, XI Latin-Iberian American Congress of Operations Research, CLAIO2002. [21] P. Hell, S. Klein, L. Tito-Nogueira, F. Protti, Partitioning chordal graphs into independent sets and cliques, Discrete Applied Mathematics (2004) 185–194. [22] P. Hell, S. Klein, L. Tito-Nogueira, F. Protti, Packing r-cliques in weighted chordal graphs, Annals of Operations Research 138 (2005) 179–187. [23] P. Horák, Q. He, W.T. Trotter, Induced matchings in cubic graphs, Journal of Graph Theory 17 (1993) 151–160. [24] C.W. Ko, F.B. Shepherd, Bipartite domination and simultaneous matroid covers, SIAM Journal on Discrete Mathematics 16 (2003) 327–349. [25] D. Kobler, U. Rotics, Finding maximum induced matchings in subclasses of claw-free and P5 -free graphs, and in graphs with matching and induced matching of equal maximum size, Algorithmica 37 (2003) 327–346. [26] László Lovász, A characterization of perfect graphs, Journal of Combinatorial Theory B 13 (1972) 95–98. [27] V.V. Lozin, On maximum induced matchings in bipartite graphs, Information Processing Letters 81 (2002) 7–11. [28] B.A. Reed, Tree width and tangles: a new connectivity measure and some applications, in: Surveys in Combinatorics, 1997 (London), in: London Math. Soc. Lecture Note Ser., vol. 241, Cambridge Univ. Press, Cambridge, 1997, pp. 87–162. [29] L.J. Stockmeyer, V.V. Vazirani, NP-completeness of some generalizations of the maximum matching problem, Information Processing Letters 15 (1982) 14–19. [30] J.R. Walter, Representations of chordal graphs as subtrees of a tree, Journal of Graph Theory 2 (1978) 265–267.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Brambles and independent packings in chordal graphsI Kathie Cameron Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
article
info
Article history: Received 31 August 2006 Accepted 28 May 2008 Available online 9 July 2008 Keywords: Independent packing Induced matching Bramble Chordal graph Min–max theorem Independent set Clique
abstract An independent packing of triangles is a set of pairwise disjoint triangles, no two of which are joined by an edge. A triangle bramble is a set of triangles, every pair of which intersect or are joined by an edge. More generally, I consider independent packings and brambles of any specified connected graphs, not just triangles. I give a min–max theorem for the maximum number of graphs in an independent packing of any family of connected graphs in a chordal graph, and a dual min–max theorem for the maximum number of graphs in a bramble in a chordal graph. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Throughout this paper, H is a fixed set of connected graphs. An H -subgraph of a given graph G is a subgraph of G isomorphic to a member of H . An H -packing of a given graph G is a pairwise node-disjoint set of H -subgraphs of G. An independent H -packing of a given graph G is an H -packing of G in which no two subgraphs of the packing are joined by an edge of G. When H is a single graph H, we have H-packings and independent H-packings of G. Thus a K2 -packing of G is precisely a matching of G, and an independent K2 -packing of G is precisely an induced matching of G. Similarly, an independent K1 -packing of G is precisely an independent set of nodes in G. The size of an independent packing is the number of graphs in the packing. The independent H -packing problem asks for the maximum number of graphs in an independent H -packing; that is, the maximum size of an independent H -packing. The problem of finding a largest induced matching, i.e., the independent K2 -packing problem, has been widely studied [6, 7,9–14,16,17,23,25,27]. It is known to be NP-hard for bipartite graphs [6,29], and for planar graphs [24]; polynomial-time algorithms for various classes of structured graphs were given in [6,7,9,11,16,17]. In [8], we showed that the independent packing problem is polynomial-time solvable for many classes of structured graphs, including weakly chordal graphs, asteroidal triple-free graphs, polygon-circle graphs, and interval-filament graphs. These classes contain other well-known classes such as chordal graphs, cocomparability graphs, circle graphs, circular-arc graphs, and outer-planar graphs. (See [4] for definitions of these classes of graphs.) Recall that a graph is called chordal if it is the intersection graph of a set of subtrees of a tree. In this paper, I give a min–max theorem for the maximum number of graphs contained in an independent H -packing of a given chordal graph G. This extends the following min–max theorem given in [6] for the maximum size of an induced matching in a chordal graph. To state the theorem, we need the following definition. A K2 -sunflower is a clique together with some incident edges. (In [6], I called these clique-neighbourhoods.) Please note that in this paper, cliques are not necessarily maximal.
I Research supported by the Natural Sciences and Engineering Research Council of Canada.
E-mail address: [email protected]. 0012-365X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2008.05.031
K. Cameron / Discrete Mathematics 309 (2009) 5766–5769
5767
Theorem 1 ([6]). For any chordal graph G, the maximum size of an induced matching in G equals the minimum number of K2 -sunflowers which together cover all the edges of G. More generally, I define an H -sunflower of a graph G to be a clique C of G together with a set P of H -subgraphs of G each of which intersects C ; C is called the centre of the sunflower and P is called the set of petals of the sunflower. The size of an H -sunflower is the number of H -subgraphs it contains. We say an H -sunflower, S, of G covers a subgraph H of G if H is a subgraph of S. The main result of this paper is the following. Theorem 2. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of H -sunflowers which together cover all the H -subgraphs of G. Theorem 2 can also be stated in the following way. Theorem 3. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of cliques of G which meet all the H -subgraphs of G. To see that Theorems 2 and 3 are equivalent, note that if S is a set of H -sunflowers of graph G which together cover all the H -subgraphs of G, then the set of centres of S is a set of cliques which meet all the H -subgraphs of G. Conversely, given a set C of cliques of G which meet all the H -subgraphs of G, if for each clique C ∈ C , we consider the sunflower S (C ) consisting of C together with all H -subgraphs of G which meet C, then {S (C ) : C ∈ C } is a set of H -sunflowers which together cover all the H -subgraphs of G. Hell, Klein, Tito-Nogueira and Protti [19–22] proved Theorem 3 in the case that H is a single fixed complete graph Kr , r ≥ 3. Given a graph G and set H of graphs, we define another graph, H (G), whose nodes are the H -subgraphs of G, and such that two nodes are adjacent exactly when the subgraphs they correspond to intersect or are joined by an edge of G. Then independent H -packings in G correspond precisely to independent sets of nodes in H (G). Where H = {K2 }, H (G) is the square of the line-graph of G. In [8], we proved the following theorem. Theorem 4 ([8]). If G is a chordal graph and H is a fixed set of connected graphs, then H (G) is a chordal graph. Chordal graphs are perfect, and from the perfect min–max theorems for chordal graphs due to Hajnal and Surányi [18] and Berge [2] together with Theorem 4, we obtain the following min–max theorems. An independent H -packing I covers an H -subgraph H if H ∈ I. Theorem 5. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of cliques of H (G) which together cover all the nodes of H (G). Theorem 6. For any chordal graph G and fixed set H of connected graphs, the maximum size of a clique in H (G) equals the minimum number of independent H -packings of G which together cover all the H -subgraphs of G. Theorems 5 and 6 will be more clear if we can characterize cliques in H (G). Reed [28] defines a bramble in graph G to be a collection of node subsets of G, each of which induces a connected subgraph of G, and every pair of which either intersect or are joined by an edge of G. Brambles are important in the study of treewidth [1,3,28]. For a fixed set H of connected graphs, I define an H -bramble in a graph G to be a set S of H -subgraphs of G, every pair of which either intersect or are joined by an edge of G. The size of the bramble is |S |. Clearly, the set of petals of any H -sunflower is an H -bramble. More generally, the set of all H -subgraphs of an H sunflower is an H -bramble. It is easy to see that H -brambles in G correspond precisely to the node-sets of cliques in H (G). An H -bramble B covers an H -subgraph H if H ∈ B . Thus Theorems 5 and 6 become the following. Theorem 7. For any chordal graph G and fixed set H of connected graphs, the maximum size of an independent H -packing in G equals the minimum number of H -brambles of G which together cover all the H -subgraphs of G. Theorem 8. For any chordal graph G and fixed set H of connected graphs, the maximum size of an H -bramble in G equals the minimum number of independent H -packings of G which together cover all the H -subgraphs of G. In order to better understand Theorems 7 and 8, I now give a characterization of H -brambles in chordal graphs. Theorem 9. Let G be a chordal graph and H a fixed set of connected graphs. Every H -bramble is contained in some H -sunflower.
5768
K. Cameron / Discrete Mathematics 309 (2009) 5766–5769
Proof. Consider a representation of G as the intersection graph of a collection F of subtrees of a tree T . Tree T can be chosen so that the nodes of T correspond to the maximal cliques in G, and so that where Tv is the subtree of T which represents v ∈ V (G), the nodes of Tv correspond to the maximal cliques of T which contain v [5,15,30]. Let N be an H -bramble, say N = {Si : 1 ≤ i ≤ k}, where each Si is an H -subgraph of G. Since subgraph Si is connected, Ti = ∪{Tv : v ∈ V (Si )} is also a subtree of T . Let F 0 = {Ti : 1 ≤ i ≤ k}. Because N is a bramble, for every i and j, 1≤ i < j ≤ k, Ti ∩ Tj 6= Φ . Then by the Helly property which holds for subtrees of a tree, ∩F 0 6= Φ . Say c ∈ ∩F 0 , where c ∈ V (T ). Then c corresponds to a maximal clique C in G, and for every i, since c ∈ ∩F 0 = ∩{Ti : 1 ≤ i ≤ k}, it follows that c ∈ Ti , and thus c ∈ Tv for some v ∈ V (Si ), and thus v ∈ C ; so C intersects each Si . Thus the graph formed by ∪N is contained in the H -sunflower consisting of C together with {Si : 1 ≤ i ≤ k}, which is contained in the H -sunflower consisting of C together with the H -subgraphs of G which intersect C . Corollary 1. In a chordal graph, every maximal H -bramble corresponds to a maximal H -sunflower, and conversely. Theorem 7 together with Corollary 1 gives Theorem 2. Theorem 8 together with Corollary 1 gives the following theorem. Theorem 10. For any chordal graph G and fixed set H of connected graphs, the maximum size of an H -sunflower equals the minimum number of independent H -packings which together cover all the H -subgraphs of G. Where H = {K2 }, Theorem 10 was proved in [6]. Given a graph G and a weighting w = (wv : v ∈ V (G)) of V (G), the weight of an independent set of nodes is the sum of the weights of the nodes in the independent set. The weight of a clique is the sum of the weights of the nodes in the clique. Given a graph G, a fixed set H of connected graphs, and a weighting w = (wH : H ∈ H ) of H , the weight of an independent H -packing is the sum of the weights of the graphs in the packing, and the weight of an H -sunflower is the sum of the weights of its H -subgraphs. Weighted versions of the perfect graph min–max theorems hold [26], and are stated below as Theorems 11 and 12. These give Theorems 13 and 14 below, which are weighted versions of Theorems 2 and 10. Theorem 11 ([26]). Let K be a perfect graph with node-set V (K ). For any integer-valued weighting w = (wv : v ∈ V (K )) of V (K ), max weight of an independent set in K
)
( X
= min
X
yC : ∀ nodes v of V (K ),
yC ≥ wv ; ∀ cliques C of K , yC ≥ 0, yC integer
.
v is in clique C
cliques C of K
Theorem 12 ([26]). Let K be a perfect graph with node-set V (K ). For any integer-valued weighting w = (wv : v ∈ V (K )) of V (K ), max weight of a clique in K
(
) X
= min
X
yI : ∀ nodes v of V (K ),
indep. sets I of K
yI ≥ wv ; ∀ indep. sets I of K , yI ≥ 0, yI integer
.
v is in indep. set I
Apply Theorem 11 to the graph K = H (G), where G is a chordal graph and where H is a fixed set of connected graphs. Then K = H (G) is a chordal graph, and thus is perfect. Also, H -subgraphs of G correspond to nodes of K = H (G); independent H -packings in G correspond to independent sets of nodes in K = H (G); and H -sunflowers of G correspond to cliques of K = H (G). Thus Theorem 11 gives the following theorem. Theorem 13. Let G be a chordal graph with node-set V (G). Let H be a fixed set of connected graphs. For any integer-valued weighting w = (wH : H ∈ H ) of H , max weight of an independent H -packing in G
= min
X
H -sunflowers S
yS : ∀H -subgraphs H of G,
X
yS ≥ wH ; ∀H -sunflowers S of G, yS ≥ 0, yS integer
H is a subgraph of sunflow er S
.
Similarly, applying Theorem 12 to the graph K = H (G), where G is a chordal graph gives the following theorem. Theorem 14. Let G be a chordal graph with node-set V (G). Let H be a fixed set of connected graphs. For any integer-valued weighting w = (wH : H ∈ H ) of H ,
K. Cameron / Discrete Mathematics 309 (2009) 5766–5769
5769
max weight of an H -sunflower in G
= min
X
indep. H -packings I
yI : ∀H -subgraphs H of G,
X H is in indep. packing I
yI ≥ wH ; ∀ indep.H -packings I of G, yI ≥ 0, yI integer
.
Where H = {K2 }, Theorems 13 and 14 were proved in [6]. References [1] Patrick Bellenbaum, Reinhard Diestel, Two short proofs concerning tree decompositions, Combinatorics, Probability and Computing 11 (2002) 541–547. [2] Claude Berge, Les problèmes de colorations en théorie des graphs, Publications Institut Statist. University Paris 9 (1960) 123–160. [3] Hans Bodlaender, Alexander Grigoriev, Arie Koster, Treewidth lower bounds with brambles, Algorithmica 51 (2008) 81–98. [4] Andreas Brandstädt, Van Bang Le, Jeremy P. Spinrad, Graph Classes - A Survey, SIAM Publications, Philadelphia, PA, 1999. [5] P. Buneman, A characterization of rigid circuit graphs, Discrete Mathematics 9 (1974) 205–212. [6] Kathie Cameron, Induced matchings, Discrete Applied Mathematics 24 (1989) 97–102. [7] Kathie Cameron, Induced matchings in intersection graphs, Discrete Mathematics 278 (2003) 1–9. [8] Kathie Cameron, Pavol Hell, Independent packings in structured graphs, Mathematical Programming Series B 105 (2006) 201–213. [9] K. Cameron, R. Sritharan, Y. Tang, Finding a maximum induced matching in weakly chordal graphs, Discrete Mathematics 266 (2003) 133–142. [10] K. Cameron, T.L. Walker, The graphs with maximum matching and maximum induced matching the same size, Discrete Mathematics 299 (2005) 49–55. [11] Jou-Ming Chang, Induced matchings in asteroidal-triple free graphs, Discrete Applied Mathematics 132 (2003) 67–78. [12] W. Duckworth, N.C. Wormald, M. Zito, Maximum induced matchings of random cubic graphs, Journal of Computational and Applied Mathematics 142 (2002) 39–50. [13] R.J. Faudree, A. Gyárfás, R.H. Schelp, Zs. Tuza, Induced matchings in bipartite graphs, Discrete Mathematics 78 (1989) 83–87. [14] G. Fricke, R.C. Laskar, Strong matchings on trees, Congressus Numerantium 89 (1992) 239–243. [15] F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, SIAM Journal on Computing 1 (1972) 47–56. [16] M.C. Golumbic, R.C. Laskar, Irredundancy in circular-arc graphs, Discrete Applied Mathematics 44 (1993) 79–89. [17] M.C. Golumbic, M. Lewenstein, New results on induced matchings, Discrete Applied Mathematics 101 (2000) 157–165. [18] A. Hajnal, J. Surányi, Über die Auflösung von Graphen in vollständige Teilgraphen, Ann. University Sci. Budapest, Eötvös, Sect. Math 1 (1958) 113–121. [19] P. Hell, S. Klein, F. Protti, L. Tito, On generalized split graphs, Electronic Notes in Discrete Mathematics 7 (2001) www.elsevier.com. [20] P. Hell, S. Klein, L.T. Nogueira, F. Protti, Independent Kr ’s in chordal graphs, XI Latin-Iberian American Congress of Operations Research, CLAIO2002. [21] P. Hell, S. Klein, L. Tito-Nogueira, F. Protti, Partitioning chordal graphs into independent sets and cliques, Discrete Applied Mathematics (2004) 185–194. [22] P. Hell, S. Klein, L. Tito-Nogueira, F. Protti, Packing r-cliques in weighted chordal graphs, Annals of Operations Research 138 (2005) 179–187. [23] P. Horák, Q. He, W.T. Trotter, Induced matchings in cubic graphs, Journal of Graph Theory 17 (1993) 151–160. [24] C.W. Ko, F.B. Shepherd, Bipartite domination and simultaneous matroid covers, SIAM Journal on Discrete Mathematics 16 (2003) 327–349. [25] D. Kobler, U. Rotics, Finding maximum induced matchings in subclasses of claw-free and P5 -free graphs, and in graphs with matching and induced matching of equal maximum size, Algorithmica 37 (2003) 327–346. [26] László Lovász, A characterization of perfect graphs, Journal of Combinatorial Theory B 13 (1972) 95–98. [27] V.V. Lozin, On maximum induced matchings in bipartite graphs, Information Processing Letters 81 (2002) 7–11. [28] B.A. Reed, Tree width and tangles: a new connectivity measure and some applications, in: Surveys in Combinatorics, 1997 (London), in: London Math. Soc. Lecture Note Ser., vol. 241, Cambridge Univ. Press, Cambridge, 1997, pp. 87–162. [29] L.J. Stockmeyer, V.V. Vazirani, NP-completeness of some generalizations of the maximum matching problem, Information Processing Letters 15 (1982) 14–19. [30] J.R. Walter, Representations of chordal graphs as subtrees of a tree, Journal of Graph Theory 2 (1978) 265–267.