On the equality of the induced matching number and the uniquely restricted matching number for subcubic graphs

TL;DR

This paper characterizes 2-connected subcubic graphs of large size where the induced matching number equals the uniquely restricted matching number, and shows these graphs can be recognized efficiently.

Contribution

It provides a complete characterization of such graphs and establishes a polynomial-time recognition algorithm.

Findings

Characterization of 2-connected subcubic graphs with equal matching numbers.

Polynomial-time recognition algorithm for these graphs.

Extension of previous work on uniquely restricted matchings.

Abstract

For a matching $M$ in a graph $G$, let $G(M)$ be the subgraph of $G$ induced by the vertices of $G$ that are incident with an edge in $M$. The matching $M$ is induced, if $G(M)$ is $1$-regular, and $M$ is uniquely restricted, if $M$ is the unique perfect matching of $G(M)$. The induced matching number $\nu_s(G)$ of $G$ is the largest size of an induced matching in $G$, and the uniquely restricted matching number $\nu_{ur}(G)$ of $G$ is the largest size of a uniquely restricted matching in $G$. Golumbic, Hirst, and Lewenstein (Uniquely restricted matchings, Algorithmica 31 (2001) 139-154) posed the problem to characterize the graphs $G$ with $\nu_s(G)=\nu_{ur}(G)$. We give a complete characterization of the $2$-connected subcubic graphs $G$ of sufficiently large order with $\nu_s(G)=\nu_{ur}(G)$. As a consequence, we are able to show that the subcubic graphs $G$ with $\nu_s(G)=\nu_{ur}(G)$ can be recognized in polynomial time.