Strong cliques in vertex‐transitive graphs

Hujdurović, Ademir

doi:10.1002/jgt.22573

Cited by 1 publication

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“…We conclude the section by noting that, as observed in [23], the equivalence between statements 1 and 2 in Theorem 6.8 cannot be generalized to arbitrary regular graphs. A small 8-regular counterexample is given by the line graph of K 6 .…”

Section: Graphs Of Small Maximum Degreementioning

confidence: 71%

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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: Graphs Of Small Maximum Degreementioning

confidence: 71%

“…A small 8-regular counterexample is given by the line graph of K 6 . Furthermore, in [23] non-localizable regular graphs are constructed in which every maximal clique is strong.…”

Section: Graphs Of Small Maximum Degreementioning

confidence: 99%