On the maximum number of maximum dissociation sets in trees with given dissociation number

In a graph G, a subset of vertices is a dissociation set if it induces a subgraph with vertex degree at most 1. A maximum dissociation set is a dissociation set of maximum cardinality. The dissociation number of G, denoted by psi (G), is the cardinality of a maximum dissociation set of G. Extremal problems involving counting the number of a given type of substructure in a graph have been a hot topic of study in extremal graph theory throughout the last few decades. In this paper, we determine the maximum number of maximum dissociation sets in a tree with prescribed dissociation number and the extremal trees achieving this maximum value. (c) 2024 Elsevier B.V. All rights reserved.