A bound on the dissociation number

Abstract

The dissociation number diss ( G ) $\text{diss}(G)$ of a graph G $G$ is the maximum order of a set of vertices of G $G$ inducing a subgraph that is of maximum degree at most 1. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graphs. For a graph G $G$ with n $n$ vertices, m $m$ edges, k $k$ components, and c 1 ${c}_{1}$ induced cycles of length 1 modulo 3, we show diss ( G ) ≥ n − 1 3 ( m + k + c 1 ) $\text{diss}(G)\ge n-\frac{1}{3}(m+k+{c}_{1})$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex‐disjoint.

Relating the independence number and the dissociation number

“…Using the notation from the proof of Claim 8, let X w w u u u u = { , , , , , } 1 2 1 1 2 2 , where C u u u u u : 1 1 2 2 1 . Similarly as in the proof of Claim 8, we obtain induced alternating paths P i and Q i between w i and some bubble B i for ∈ i [2]. First, suppose that u 1 and u 2 are adjacent.…”

“…The dissociation number is algorithmically hard even when restricted, for instance, to subcubic bipartite graphs [3,14,18]. Bounds [2,4,5,9], fast exact algorithms [13], (randomized) approximation algorithms [12,13], fixed parameter tractability [16], and the maximum number of maximum dissociation sets [17] have been studied for this parameter or its dual, the 3-path (vertex) cover number.…”