Relating dissociation, independence, and matchings

TL;DR

This paper investigates the dissociation number in graphs, providing efficient recognition of certain cases, bounds, and NP-hardness results for computing it, with implications for approximation algorithms in bipartite graphs.

Contribution

It characterizes when the inequality for dissociation number bounds holds with equality and shows how to efficiently find maximum dissociation sets in those cases.

Findings

Recognition of equality cases for dissociation bounds is efficient.

Bounds on dissociation number relate to independence and induced matching numbers.

Deciding equality with bounds is NP-hard.

Abstract

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.