### 1-well-covered graphs containing a clique of size n∕3

#### Abstract

A graph is well-covered if all of its maximal independent sets have the same size. A graph that remains well-covered upon the removal of any vertex is called a 1-well-covered graph. These graphs, when they have no isolated vertices, are also known as W_{2} graphs. It is well-known that every graph G ∈ W_{2} has two disjoint maximum independent sets. In this paper, we investigate connected W_{2} graphs with n vertices that contain a clique of size n∕3. We prove that if the removal of two disjoint maximum independent sets from a graph G ∈ W_{2} leaves a clique of size at least 3, then G contains a clique of size n∕3. Using this result, we provide a complete characterization of these graphs, based on eleven graph families.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2024.12.2.8

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