On the Maximal Number of Maximum Dissociation Sets in Forests with Fixed Order and Dissociation Number

Abstract

Given a graph $G$ with $S \subseteq V_G$, we call $S$ a maximum dissociation set if the induced subgraph $G[S]$ contains no path of order $3$, and subject to this condition, the subset $S$ has the maximum cardinality. The dissociation number of $G$ is the cardinality of a maximum dissociation set. Inspired by the results of [26, 27] on the maximal number of maximum dissociation sets, in this contribution we investigate the maximal number of maximum dissociation sets in forests with fixed order and dissociation number. Firstly, a lower bound on the dissociation number of a forest with fixed order is established, and all extremal graphs are determined. Secondly, all trees (resp. forests) having the largest and the second largest number of maximum dissociation sets among trees (resp. forests) with given order and dissociation number are completely characterized. Finally, we show that the results in [26, 27] can be deduced by our results.