A note on the second-largest number of dissociation sets in connected graphs

TL;DR

The paper addresses the problem of identifying the second-largest number of dissociation sets among connected graphs of order $n$, building on prior work that determined the maximum and its extremal graphs. It develops a framework of exact counting via the recurrence for $d(G)$, leverages structural lemmas on trees and unicyclic graphs, and uses the $K_r\ast G$ construction to describe extremal configurations. The main contributions are a complete determination that, for $n \ge 9$, the second-largest dissociation-set count is achieved by the graphs $U_n$ (unicyclic) and $T_n$ (trees) with explicit bounds $h(n)$, and the refinement that the second-largest among trees is characterized by $T_n$ with specified structure; together, these results resolve the open question and provide precise extremal graphs. The findings advance extremal dissociation-set theory and offer exact counts and constructions that can inform related enumeration problems in graph substructures.

Abstract

A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order n and characterized the corresponding extremal graphs.They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order n and the corresponding extremal graphs. In this paper, we give a positive answer to this question.

Figures (10)

• Figure 1: $K_{r} \ast (K_{s_{1}}\cup K_{s_{2}}\cup \cdots \cup K_{s_{t}})$.
• Figure 2: $T$, ${T}'$, ${T}"$.
• Figure 3: Structure of $T$ in Claim 1.
• Figure 4: Structure of $T$ in Claim 2.
• Figure 5: The two possible structures of $T$ if $d(T-v_0)=f(n-1)$.

Theorems & Definitions (10)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 2.6
• Corollary 2.7
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3