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arXiv · 1609.06961

Graphs vertex-partitionable into strong cliques

Ademir Hujdurović·Martin Milanič·Bernard Ries
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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 $k$-partite well-covered graphs having all maximal cliques of size $k$. Our results unify and generalize several results from the literature.

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Ademir Hujdurović, Martin Milanič, Bernard Ries. 2017-01-06. Graphs vertex-partitionable into strong cliques. https://arxiv.org/abs/1609.06961

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