Clique-free t-matchings in degree-bounded graphs

TL;DR

This paper introduces fast combinatorial algorithms for finding maximum size or weight t-matchings in degree-bounded graphs that avoid certain forbidden subgraphs, generalizing known problems like triangle-free matchings.

Contribution

It presents the first algorithms for weighted versions and faster algorithms for unweighted cases of clique-free t-matchings in degree-bounded graphs.

Findings

Algorithms are simple and fast, improving previous methods.

First algorithms for weighted clique-free t-matchings.

Applicable to general forbidden subgraph variants.

Abstract

We consider problems of finding a maximum size/weight $t$-matching without forbidden subgraphs in an undirected graph $G$ with the maximum degree bounded by $t+1$, where $t$ is an integer greater than $2$. Depending on the variant forbidden subgraphs denote certain subsets of $t$-regular complete partite subgraphs of $G$. A graph is complete partite if there exists a partition of its vertex set such that every pair of vertices from different sets is connected by an edge and vertices from the same set form an independent set. A clique $K_t$ and a bipartite clique $K_{t,t}$ are examples of complete partite graphs. These problems are natural generalizations of the triangle-free and square-free $2$-matching problems in subcubic graphs. In the weighted setting we assume that the weights of edges of $G$ are vertex-induced on every forbidden subgraph. We present simple and fast combinatorial algorithms for these problems. The presented algorithms are the first ones for the weighted versions, and for the unweighted ones, are faster than those known previously. Our approach relies on the use of gadgets with so-called half-edges. A half-edge of edge $e$ is, informally speaking, a half of $e$ containing exactly one of its endpoints.