Efficient Enumeration of Induced Matchings in a Graph without Cycles with Length Four

TL;DR

This paper presents a new efficient algorithm for enumerating induced matchings in graphs without 4-cycles, achieving constant time per solution in such graphs and exploring fixed parameter tractability for broader classes.

Contribution

It introduces a novel $O(1)$ time complexity algorithm for induced matching enumeration in $C_4$-free graphs and analyzes fixed parameter tractability for related graph classes.

Findings

Constant time enumeration in $C_4$-free graphs

Fixed parameter tractability for bounded tree-width graphs

Enumeration efficiency in graphs without 4-cycles

Abstract

We address the induced matching enumeration problem. An edge set $M$ is an induced matching of a graph $G =(V,E)$. The enumeration of matchings are widely studied in literature, but the induced matching has not been paid much attention. A straightforward algorithm takes $O(|V|)$ time for each solution, that is coming from the time to generate a subproblem. We investigated local structures that enables us to generate subproblems in short time, and proved that the time complexity will be $O(1)$ if the input graph is $C_4$-free. A $C_4$-free graph is a graph any whose subgraph is not a cycle of length four. Finally, we show the fixed parameter tractability of counting induced matchings for graphs with bounded tree-width and planar graphs.