  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 ${\rm diss}(G)\leq\frac{4}{3}\alpha(G-M)$ implicit in
their work, where $G$ is a bipartite graph, $M$ is a maximum matching in $G$,
and $\alpha(G-M)$ 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 $\max\{
\alpha(G),2\nu_s(G)\} \leq {\rm diss}(G)\leq \alpha(G)+\nu_s(G)\leq
2\alpha(G)$, where $\nu_s(G)$ 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.
