On graphs having maximal independent sets of exactly $t$ distinct cardinalities

TL;DR

This paper characterizes graphs with no degree-one vertices that have exactly t distinct sizes of maximal independent sets, focusing on cases where t is at least four and the shortest cycle length is sufficiently large.

Contribution

It provides a complete characterization of such graphs under specified conditions for t ≥ 4 and large shortest cycle length.

Findings

Graphs with the given properties are fully characterized for t ≥ 4 and shortest cycle length ≥ 6t-6.

The study identifies structural constraints on these graphs.

Results contribute to understanding the relationship between cycle length and independent set cardinalities.

Abstract

For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and the length of the shortest cycle is at least 6t-6, we completely characterize such graphs.