Upper bound for the number of maximal dissociation sets in trees

Abstract

Let $G$ be a simple graph. A dissociation set of $G$ is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most 1. A dissociation set is maximal if it is not contained as a proper subset in any other dissociation set. We introduce the notation $Φ(G)$ to represent the number of maximal dissociation sets in $G$. This study focuses on trees, specifically showing that for any tree $T$ of order $n\geq4$, the following inequality holds: \[Φ(T)\leq 3^{\frac{n-1}{3}}+\frac{n-1}{3}.\] We also identify the extremal tree that attains this upper bound. Additionally, to establish the upper bound on the number of maximal dissociation sets in trees of order $n$, we also determine the second largest number of maximal dissociation sets in forests of order $n$.

Figures (19)

• Figure 1: $T^*_n$
• Figure 2: $T^*_{8}$
• Figure 3: $T'=T-uv+ux$
• Figure 4: $T'=T-xt+xy$
• Figure 5: The support vertex $v$ is adjacent to $k\geq3$ leaves.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Claim 1
• Claim 2
• Claim 3
• Conjecture 1