A bound on the dissociation number

Abstract

The dissociation number $\mathrm{diss}(G)$ 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 $\mathrm{diss}(G)\ge n-\frac{1}{3}(m+k+c_1)$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex-disjoint.