Graphs with minimal well-covered dimension

TL;DR

This paper investigates the well-covered dimension of certain graph classes, showing that some graphs, including chordal and Sierpinski gasket graphs, have well-covered dimension equal to their simplicial clique number, generalizing previous results.

Contribution

It introduces a class of graphs with well-covered dimension equal to the simplicial clique number, extending known results to new graph families including Sierpinski gasket graphs.

Findings

Graphs with well-covered dimension equal to the simplicial clique number include chordal and certain other graphs.

All Sierpinski gasket graphs of order at least 2 have well-covered dimension 3.

The class of graphs studied generalizes previous results on chordal graphs.

Abstract

There is a class of graphs with well-covered dimension equal to the simplicial clique number that contains all chordal graphs and infinitely many other graphs. These graphs generalize a result by Brown and Nowakowski on the well-covered dimension of chordal graphs. Furthermore, each member of the infinite family of Sierpinski gasket graphs of order at least $2$ has well-covered dimension $3,$ the simplicial clique number.