Well covered simplicial, chordal, and circular arc graphs

Prisner, Erich; Topp, Jerzy; Vestergaard, Preben Dahl

doi:10.1002/(sici)1097-0118(199602)21:2<113::aid-jgt1>3.0.co;2-u

Cited by 47 publications

(37 citation statements)

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“…Thus, Theorem 3.1 yields the following. Recall that, as noted in Table 1, it follows from results of Prisner et al [36] that localizable graphs can be recognized in polynomial time within the class of chordal graphs (and, more generally, also in the class of C 4 -free graphs [24]).…”

Section: A Hardness Proof and Its Implicationsmentioning

confidence: 95%

Detecting strong cliques

Hujdurović

Milanič

Ries

2019

Discrete Mathematics

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A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their complexity in the classes of chordal graphs, weakly chordal graphs, line graphs and their complements, and graphs of maximum degree at most three. Our results rely on connections with matchings and relate to several graph properties studied in the literature, including well-covered graphs, localizable graphs, and general partition graphs.In this paper we report on advances regarding the computational complexity of several problems related to strong cliques in graphs. We pay particular attention to the class of graphs whose vertex set can be partitioned into strong cliques. Such graphs were called localizable by Yamashita and Kameda in 1999 [41] and studied further by Hujdurović et al. in 2018 [24]. Localizable graphs form a rich class of well-covered graphs, that is, graphs in which all maximal independent sets are of the same size. Moreover, localizable graphs and well-covered graphs coincide within the class of perfect graphs (see, e.g., [24]). Well-covered graphs were introduced by Plummer in 1970 [34] and have been studied extensively in the literature (see, e.g., the surveys by Plummer [35] and Hartnell [20]).Relatively few complexity results regarding problems involving strong cliques are available in the literature. Zang [42] showed that it is co-NP-complete to test whether a given clique in a graph is strong and Hoàng [22] established NP-hardness of testing whether a given graph contains a strong clique. Hujdurović et al. [24] showed that recognizing localizable graphs is NP-hard, even in the class of well-covered graphs whose complements are localizable. Polynomial-time recognition algorithms for localizable graphs are known within several classes of graphs, including triangle-free graphs, C 4 -free graphs, and line graphs (see [24, Section 4.1] for an overview). Results of Hoàng [21] and Burlet and Fonlupt [10] imply the existence of a polynomial-time algorithm for determining if in every induced subgraph of a given graph G, each vertex belongs to a strong clique. On the other hand, to the best of our knowledge, the complexity status of recognizing several graph classes related to strong cliques is in general still open. In particular, this is the case for: (i) the problem of recognizing strongly perfect graphs [4,5], introduced by Berge as graphs every induced subgraph of which has a strong independent set, (ii) the problem of determining whether every edge of a given graph is contained in a strong clique or, equivalently, the problem of recognizing general partition graphs (see, e.g., [25,30,32]), and (iii) the problem of recognizing CIS graphs, defined as graphs in which every maximal clique is strong, or, equivalently, as graphs in which every maximal independent set is strong (see, e.g., [7,9,15,19,40]).In this paper we report on several advances regarding the comput...

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Section: A Hardness Proof and Its Implicationsmentioning

confidence: 95%

Detecting strong cliques

Hujdurović

Milanič

Ries

2019

Discrete Mathematics

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“…Recall that a simplicial vertex is one whose neighborhood is complete. Prisner, Topp, and Vestergaard [5] considered simplicial vertices in well-covered graphs. In particular, they defined a simplex as a maximal clique containing a simplicial vertex, and showed that if a graph is well-covered then the simplices are vertex-disjoint, and if every vertex belongs to exactly one simplex then the graph is well-covered.…”

Section: Triangles and Simplicial Verticesmentioning

confidence: 99%

Graphs in which all maximal bipartite subgraphs have the same order

2020

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Motivated by the concept of well-covered graphs, we define a graph to be wellbicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with wellcovered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bicovered. arXiv:1909.07503v1 [math.CO]

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“…9 if S * is not independent then 10 return ∅. 11 Construct the graph H * x . 12 Construct the flow network F…”

Section: Algorithmmentioning

confidence: 99%

Recognizing generating subgraphs in graphs without cycles of lengths 6 and 7

Tankus

2020

Discrete Applied Mathematics

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Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition BX and BY . The subgraph B is generating if there exists an independent set S such that each of S ∪ BX and S ∪ BY is a maximal independent set in the graph. If B is generating, it produces the restriction w(BX ) = w(BY ).Let w : V (G) −→ R be a weight function. We say that G is w-well-covered if all maximal independent sets are of the same weight. The graph G is w-well-covered if and only if w satisfies all restrictions produced by all generating subgraphs of G. Therefore, generating subgraphs play an important role in characterizing weighted well-covered graphs.It is an NP-complete problem to decide whether a subgraph is generating, even when the subgraph is isomorphic to K1,1 [1]. We present a polynomial algorithm for recognizing generating subgraphs for graphs without cycles of lengths 6 and 7.

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Well covered simplicial, chordal, and circular arc graphs

Abstract: A graph G is called well covered if every two maximal independent sets of G have the same number of vertices. In this paper, we characterize well covered simplicial, chordal and circular arc graphs. © 1996 John Wiley & Sons, Inc.

Cited by 47 publications

References 0 publications

Detecting strong cliques

Detecting strong cliques

Graphs in which all maximal bipartite subgraphs have the same order

Recognizing generating subgraphs in graphs without cycles of lengths 6 and 7

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