A simple combinatorial algorithm for restricted 2-matchings in subcubic graphs - via half-edges

Abstract

We consider three variants of the problem of finding a maximum weight restricted 2-matching in a subcubic graph G. (A 2-matching is any subset of the edges such that each vertex is incident to at most two of its edges.) Depending on the variant a restricted 2-matching means a 2-matching that is either triangle-free or square-free or both triangle- and square-free. Since computing a maximum weight square-free 2-matching in a subcubic graph is NP-hard, in the second and third variant we additionally assume that the edge-weights are vertex-induced on each square. While there exist polynomial time algorithms for the first two types of 2-matchings, they are quite complicated or use advanced methodology. For each of the three problems we present a simple reduction to the computation of a maximum weight b-matching. The reduction is conducted with the aid of half-edges. A half-edge of edge e is, informally speaking, a half of e containing exactly one of its endpoints. For a subset of triangles and/or squares of G, we replace each edge of such a triangle/square with two half-edges. Two half-edges of one edge e of weight w(e) may get different weights, not necessarily equal to 12w(e). In the metric setting when the edge weights satisfy the triangle inequality, this has a geometric interpretation connected to how an incircle partitions the edges of a triangle. Our algorithms are additionally faster than those known before. The running time of each of them is O(n2log⁡n), where n denotes the number of vertices in the graph.