Graphs vertex-partitionable into strong cliques

Abstract

A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs.In this paper we obtain several structural and algorithmic results about localizable graphs. Our results include a proof of the fact that every very well-covered graph is localizable and characterizations of localizable graphs within the classes of line graphs, triangle-free graphs, C 4 -free graphs, and cubic graphs, each leading to a polynomial time recognition algorithm. On the negative side, we prove NP-hardness of recognizing localizable graphs within the classes of weakly chordal graphs, complements of line graphs, and graphs of independence number three. Furthermore, using localizable graphs we disprove a conjecture due to Zaare-Nahandi about kpartite well-covered graphs having all maximal cliques of size k. Our results unify and generalize several results from the literature.

Detecting strong cliques

“…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]).…”

“…Replacing the graph G with its complement G maps the strong cliques of G into the equivalent concept of strong independent sets of G, that is, independent sets intersecting every maximal clique in G. The notions of strong cliques and strong independent sets in graphs played an important role in the study of perfect graphs and their subclasses (see, e.g., [4,5,10,21,31]). Moreover, various other graph classes studied in the literature can be defined in terms of properties involving strong cliques (see, e.g., [7,8,15,24,25,30]). In some cases, strong cliques can be seen as a generalization of perfect matchings: transforming any regular triangle-free graph to the complement of its line graph maps any perfect matching into a strong clique (see [6,33] for applications of this observation).…”

“…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]).…”

“…On the other hand, to the best of our knowledge, the complexity of the remaining three problems has not been determined in the literature. One of the aims of this paper is to partially close this gap and, more generally, to obtain further complexity and algorithmic results on the six strong clique problems in specific classes of graphs.Our work is also motivated by the general quest for finding further applications of connections between strong cliques and other graph theoretic concepts, in particular: (i) of the fact that localizable graphs and well-covered graphs coincide within the class of perfect graphs, as observed in [24], and (ii) of connections between strong cliques and matchingswhich naturally appear in the study of line graphs and their complements [6,8,24,27,33].Regarding (i), we further elaborate on the approach used by Chvátal and Slater [14] and Sankaranarayana and Stewart [39] for proving hardness of recognizing well-covered graphs within the class of weakly chordal graphs (which are perfect), to derive similar results for Strong Clique Vertex Cover and Strong Clique Partition, two of the three problems from the above list whose complexity status was unknown prior to this work. These results are presented in Section 3.Many of our results motivated by (ii) are based on the study of Strong Clique Extension, an auxiliary problem which we introduce in Section 4.…”

A Characterization of Claw-free CIS Graphs and New Results on the Order of CIS Graphs

“…Hujdurović et al [16] showed that a clique in a C 4 -free graph is strong if and only if it is simplicial, which leads to a polynomially testable characterization of CIS C 4 -free graphs. The concept of strong clique gives rise to several other interesting graph properties studied in the literature (see, e.g., [7,16,17,24]).…”