The minimum number of maximal dissociation sets in unicyclic graphs

TL;DR

This paper establishes a lower bound on the number of maximal dissociation sets in unicyclic graphs and characterizes the graphs that achieve this minimum, advancing understanding of graph dissociation properties.

Contribution

It proves that every unicyclic graph with at least 3 vertices has at least loor n/2+2 maximal dissociation sets and identifies the extremal graphs.

Findings

Minimum number of maximal dissociation sets is loor n/2+2 for unicyclic graphs.

Characterization of graphs that attain this minimum bound.

Provides insights into dissociation set structure in unicyclic graphs.

Abstract

A subset of vertices in a graph $G$ is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for $n\geq 3$, every unicyclic graph contains a minimum of $\lfloor n/2\rfloor+2$ maximal dissociation sets. We also show the graphs that attain this minimum bound.