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

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