The maximum number of maximum dissociation sets in potted graphs

Abstract

A potted graph is a unicyclic graph such that its cycle contains a unique vertex with degree larger than 2. Given a graph G, a subset of $V\left(G\right)$ is a dissociation set of G if it induces a subgraph with maximum degree at most one. A maximum dissociation set is a dissociation set with maximum cardinality. In this paper, we determine the maximum number of maximum dissociation sets in a potted graph of order n which contains a fixed cycle. The corresponding extremal graphs are also characterized.