Well-Covered Graphs Without Cycles of Lengths 4, 5 and 6

TL;DR

This paper characterizes the space of weight functions making certain graphs well-covered, focusing on graphs without cycles of lengths 4, 5, and 6, and provides polynomial algorithms for recognizing specific subgraph structures.

Contribution

It offers a polynomial characterization of weight functions for well-covered graphs without cycles of lengths 4, 5, and 6, and develops algorithms for recognizing generating subgraphs and relating edges.

Findings

Polynomial characterization of w-well-covered graphs without 4, 5, and 6 cycles.

Polynomial algorithms for recognizing generating subgraphs in graphs without cycles of lengths 5, 6, and 7.

Polynomial algorithms for recognizing relating edges in graphs without cycles of lengths 5 and 6.

Abstract

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of length 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition B_{X} and B_{Y}. Assume that there exists an independent set S such that both the union of S and B_{X} and the union of S and B_{Y} are maximal independent sets of G. Then B is a generating subgraph of G, and it produces the restriction w(B_{X})=w(B_{Y}). It is known that for every weight function w, if G is w-well-covered, then the above restriction is satisfied. In the special case, where B_{X}={x} and B_{Y}={y}, we say that xy is a relating edge. Recognizing relating edges and generating subgraphs is an NP-complete problem. However, we provide a polynomial algorithm for recognizing generating subgraphs of an input graph without cycles of length 5, 6 and 7. We also present a polynomial algorithm for recognizing relating edges in an input graph without cycles of length 5 and 6.