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

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. 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 of \(G\), we replace each edge of such a triangle with two half-edges. Two half-edges of one edge \(e\) of weight \(w(e)\) may get different weights, not necessarily equal to \(\frac{1}{2}w(e)\). 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 \(O(n^2\log{n})\), where \(n\) denotes the number of vertices in the graph.