A bound on the dissociation number

TL;DR

This paper establishes a lower bound on the dissociation number of graphs based on their structural parameters and characterizes extremal graphs where cycles are vertex-disjoint.

Contribution

It provides a new lower bound for the dissociation number and characterizes extremal graphs with disjoint cycles, advancing understanding of graph dissociation properties.

Findings

Lower bound: diss(G) ≥ n - (1/3)(m + k + c_1)

Characterization of extremal graphs with vertex-disjoint cycles

Applicable to graphs with complex cycle structures

Abstract

The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ 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 $G$ with $n$ vertices, $m$ edges, $k$ components, and $c_1$ induced cycles of length $1$ modulo $3$, we show ${\rm diss}(G)\geq n-\frac{1}{3}\Big(m+k+c_1\Big)$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex-disjoint.