# 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 1/2w(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^2logn), where n denotes the number of vertices in the graph.

## Authors

• 4 publications
• 1 publication
• ### Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

We show that triangle-free penny graphs have degeneracy at most two, lis...
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• ### Maximum Weight Independent Sets for (S_1,2,4,Triangle)-Free Graphs in Polynomial Time

The Maximum Weight Independent Set (MWIS) problem on finite undirected g...
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• ### Weighted Triangle-free 2-matching Problem with Edge-disjoint Forbidden Triangles

The weighted T-free 2-matching problem is the following problem: given a...
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• ### Maximum independent sets in (pyramid, even hole)-free graphs

A hole in a graph is an induced cycle with at least 4 vertices. A graph ...
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• ### Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

Triangle counting is a fundamental technique in network analysis, that h...
06/05/2021 ∙ by Noujan Pashanasangi, et al. ∙ 0

• ### Minimum Area All-flush Triangles Circumscribing a Convex Polygon

In this paper, we consider the problem of computing the minimum area tri...
12/14/2017 ∙ by Kai Jin, et al. ∙ 0

• ### An algorithm for reconstruction of triangle-free linear dynamic networks with verification of correctness

Reconstructing a network of dynamic systems from observational data is a...
03/05/2020 ∙ by Mihaela Dimovska, et al. ∙ 0

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