A bound on the dissociation number - Bock, Felix; Pardey, Johannes; Penso, Lucia D.; Rautenbach, Dieter (Journal of graph theory, 2023)
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.