Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Brandstädt, Andreas; Giakoumakis, Vassilis; Maffray, Frédéric

doi:10.1016/j.dam.2011.10.031

Cited by 24 publications

(15 citation statements)

References 39 publications

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“…, 6 we define sets A i , A + i , A − i , and Q, with respect to G, v and H, as in the last paragraph of Section 2. Then we have the following properties: 4 , v 5 , x, z} induces a graph which is isomorphic to H 4 in G, a contradiction to Lemma 1. So, A + 2 = ∅, and the claim holds.…”

Section: (Smentioning

confidence: 89%

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Independent Sets in Classes Related to Chair-Free Graphs

Karthick

2016

Algorithms and Discrete Applied Mathematics

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The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. MWIS is known to be N P -complete in general, even under various restrictions. Let S i,j,k be the graph consisting of three induced paths of lengths i, j, k with a common initial vertex. The complexity of the MWIS problem for S 1,2,2 -free graphs, and for S 1,1,3 -free graphs are open. In this paper, we show that the MWIS problem can solved in polynomial time for (S 1,2,2 , S 1,1,3 , cochair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This extends some known results in the literature.

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Section: (Smentioning

confidence: 89%

“…∈ E, and then since {x, a 1 , b 1 , p, t} does not induce a co-chair in G, xp ∈ E. Then since {x, b 1 , p, a 2 , q} does not induce a co-chair in G, xq ∈ E. But, now 4 , p} induces an S 1,1,3 in G, which is a contradiction. Thus, G is H 6 -free.…”

Section: Introductionmentioning

confidence: 93%

Independent Sets in Classes Related to Chair-Free Graphs

Karthick

2016

Algorithms and Discrete Applied Mathematics

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“…Finally, this work furthers the knowledge about diamond-free graphs. In 1984, Tucker proved the Strong Perfect Graph Conjecture, now Strong Perfect Graph Theorem, for diamond-free graphs [42], and more recently there has been an increased interest regarding the coloring problem and the chromatic number of diamond-free graphs and their subclasses (see, e.g., [11,17,29,31]). Diamond-free graphs also played an important role in a recent work of Chudnovsky et al [15] who proved that there are exactly 24 4-critical P 6 -free graphs.…”

Section: Introductionmentioning

confidence: 99%

Strong cliques in diamond-free graphs

Chiarelli¹,

Barona²,

Milanič³

et al. 2020

Preprint

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A strong clique in a graph is a clique intersecting all inclusion-maximal stable sets. Strong cliques play an important role in the study of perfect graphs. We study strong cliques in the class of diamond-free graphs, from both structural and algorithmic points of view. We show that the following five NP-hard or co-NP-hard problems remain intractable when restricted to the class of diamond-free graphs: Is a given clique strong? Does the graph have a strong clique? Is every vertex contained in a strong clique? Given a partition of the vertex set into cliques, is every clique in the partition strong? Can the vertex set be partitioned into strong cliques? On the positive side, we show that the following two problems whose computational complexity is open in general can be solved in linear time in the class of diamond-free graphs: Is every maximal clique strong? Is every edge contained in a strong clique? These results are derived from a characterization of diamond-free graphs in which every maximal clique is strong, which also implies an improved Erdős-Hajnal property for such graphs.

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“…Using the approach of [5,6], Basavaraju, Chandran and Karthick in [1] showed that the MWIS problem can be solved in polynomial time for hole-and dart-free graphs by reducing it first to hole-, dart-, and gem-free graphs (which are hole-and paraglider-free). We extend previous results and show:…”

Section: Introductionmentioning

confidence: 99%

“…In [6], it was shown that the atoms of the graphs in the class (i) are weakly chordal or specific graphs, and in [5], it was shown that the prime atoms of the graphs in the class (ii) are nearly weakly chordal, from which polynomial time algorithms for MWIS on these graphs follow.…”

Section: Introductionmentioning

confidence: 99%

Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull

Brandstädt

Mosca

2014

Graphs and Combinatorics

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The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying clique separator decomposition as well as modular decomposition, we obtain polynomial time solutions of MWIS for oddhole-and dart-free graphs as well as for odd-hole-and bull-free graphs (dart and bull have five vertices, say a, b, c, d, e, and dart has edges ab, ac, ad, bd, cd, de, while bull has edges ab, bc, cd, be, ce). If the graphs are hole-free instead of odd-hole-free then stronger structural results and better time bounds are obtained.

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Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Cited by 24 publications

References 39 publications

Independent Sets in Classes Related to Chair-Free Graphs

Independent Sets in Classes Related to Chair-Free Graphs

Strong cliques in diamond-free graphs

Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull

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