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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Deniz, Zakir
Duzce University
Electrical & Electronics Engineering

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1-well-covered graphs containing a clique of size n∕3 Deniz, Zakir
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 12, No 2 (2024): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2024.12.2.8

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 W2 graphs. It is well-known that every graph G ∈ W2 has two disjoint maximum independent sets. In this paper, we investigate connected W2 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 ∈ W2 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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