Relating dissociation, independence, and matchings

A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most $1$. Computing the dissociation number ${\rm diss}(G)$ of a given graph $G$, defined as the order of a maximum dissociation set in $G$, is algorithmically hard even when $G$ is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple $\frac{4}{3}$-approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality ${\rm diss}(G)\leq\frac{4}{3}α(G-M)$ implicit in their work, where $G$ is a bipartite graph, $M$ is a maximum matching in $G$, and $α(G-M)$ denotes the independence number of $G-M$. We show that the pairs $(G,M)$ for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph $G$ satisfies $\max\{ α(G),2ν_s(G)\} \leq {\rm diss}(G)\leq α(G)+ν_s(G)\leq 2α(G)$, where $ν_s(G)$ denotes the induced matching number of $G$. We show that deciding whether ${\rm diss}(G)$ equals any of the four terms lower and upper bounding ${\rm diss}(G)$ is NP-hard.