Weighted Well-Covered Graphs without Cycles of Length 4, 5, 6 and 7

TL;DR

This paper studies weighted well-covered graphs, proving that for graphs without cycles of lengths 4 to 7, the set of weight functions making the graph w-well-covered can be computed efficiently.

Contribution

It introduces a polynomial-time algorithm to find the weight function space for w-well-covered graphs lacking certain cycle lengths.

Findings

Polynomial-time computation of weight function space for specific graphs

Characterization of well-covered graphs without small cycles

Extension of recognition results to weighted cases

Abstract

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove that finding the vector space of weight functions under which an input graph is w-well-covered can be done in polynomial time, if the input graph does not contain cycles of length 4, 5, 6 and 7.