1-well-covered graphs revisited

TL;DR

This paper explores properties and characterizations of 1-well-covered graphs, a subclass of well-covered graphs, and investigates how certain graph operations affect their structure and well-coveredness.

Contribution

It provides new characterizations of 1-well-covered graphs and examines how graph operations like corona, join, and concatenation influence their properties.

Findings

Deleting a shedding vertex preserves well-coveredness.

G belongs to W_{2} if and only if it is 1-well-covered without isolated vertices.

New characterizations of 1-well-covered graphs are introduced.

Abstract

A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples, 1975). A graph G belongs to class W_{n} if every n pairwise disjoint independent sets in G are included in $n$ pairwise disjoint maximum independent sets (J. W. Staples, 1975). Clearly, W_{1} is the family of all well-covered graphs. It turns out that G belongs to W_{2} if and only if it is a 1-well-covered graph without isolated vertices. We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given independent set A in a graph G. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G-v is well-covered. In addition, we provide new characterizations of 1-well-covered graphs, which we further use in building 1-well-covered graphs by corona, join, and concatenation operations.