The number of dissociation sets in connected graphs

TL;DR

This paper investigates the maximum number of dissociation sets in connected graphs, providing exact formulas for extremal cases, and characterizes the structures that achieve these bounds.

Contribution

It introduces methods to increase dissociation sets and determines the maximum counts for connected graphs and trees, identifying extremal structures.

Findings

Maximum dissociation sets in connected graphs are given by explicit formulas.

Characterization of extremal graphs achieving maximum dissociation sets.

Identification of the second-best unicyclic graph for dissociation sets.

Abstract

Extremal problems related to the enumeration of graph substructures, such as independent sets, matchings, and induced matchings, have become a prominent area of research with the advancement of graph theory. A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most $1$, making it a natural generalization of these previously studied substructures. In this paper, we present efficient tools to strictly increase the number of dissociation sets in a connected graph. Furthermore, we establish that the maximum number of dissociation sets among all connected graphs of order $n$ is given by \begin{align*} \begin{cases} 2^{n-1}+(n+3)\cdot 2^{\frac{n-5}{2}}, &~ {\rm if}~ n~{\rm is}~{\rm odd};\\ 2^{n-1}+(n+6)\cdot 2^{\frac{n-6}{2}}, &~ {\rm if}~ n~{\rm is}~{\rm even}. \end{cases} \end{align*} Additionally, we determine the achievable upper bound on the number of dissociation sets in a tree of order $n$ and characterize the corresponding extremal graphs as an intermediate result. Finally, we identify the unicyclic graph that is the candidate for having the second largest number of dissociation sets among all connected graphs.