The minimum number of maximal dissociation sets 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, we show that every unicyclic graph of order $n$ with $n\ge 3$ contains at least $\lfloor \frac{n}{2}\rfloor +2$ maximal dissociation sets. We also characterize the graphs attaining this lower bound.