On the recognition of k-equistable graphs

Authors:
Vadim E. Levit;Martin Milanič;David Tankus
Affiliations:
Department of Computer Science and Mathematics, Ariel University Center of Samaria, Ariel, Israel;University of Primorska, UP IAM, Koper, Slovenia,University of Primorska, UP FAMNIT, Koper, Slovenia;Department of Computer Science and Mathematics, Ariel University Center of Samaria, Ariel, Israel
Venue:
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Year:
2012

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Abstract

A graph G=(V,E) is called equistable if there exist a positive integer t and a weight function $w:V \longrightarrow \mathbb{N}$ such that S⊆V is a maximal stable set of G if and only if w(S)=t. The function w, if exists, is called an equistable function of G. No combinatorial characterization of equistable graphs is known, and the complexity status of recognizing equistable graphs is open. It is not even known whether recognizing equistable graphs is in NP. Let k be a positive integer. An equistable graph G=(V,E) is said to be k-equistable if it admits an equistable function which is bounded by k. For every constant k, we present a polynomial time algorithm which decides whether an input graph is k-equistable.