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

Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

“…First, for subcubic graphs, i.e., graphs with maximum degree at most three, polynomial-time algorithms for the C 4 -free and the C ≤4 -free 2-matching problems were given by Bérczi and Kobayashi [3] and Bérczi and Végh [4], respectively. Simpler algorithms for both problems in subcubic graphs (and for some of their weighted variants) were designed by Hartvigsen and Li [16] and by Paluch and Wasylkiewicz [36]. It is worth noting that a connection between the C 4 -free matching problem and a connectivity augmentation problem is highlighted in [3], underscoring the significance of the C 4 -free matching problem.…”

“…It is shown by Király (see [13]) and by Bérczi and Kobayashi [3] that the weighted C ≤4 -free 2-matching problem is NP-hard even if the input graph is restricted to be cubic, bipartite, and planar. For the weighted C 4 -free 2-matching problem in bipartite graphs, and more generally for the weighted K t,t -free t-matching problem in bipartite graphs, under the assumption that the weight function satisfies a certain property, Makai [31] gave a polyhedral description, Takazawa [44] designed a combinatorial polynomial-time algorithm, and Paluch and Wasylkiewicz [35] presented a faster and simpler algorithm.…”

“…It is still open whether the weighted C 3 -free 2-matching problem can be solved in polynomial time. For the weighted C 3 -free 2-matching problem in subcubic graphs, Hartvigsen and Li [17] gave a polyhedral description and a polynomial-time algorithm, and faster polynomial-time algorithms were presented by Kobayashi [21] and by Paluch and Wasylkiewicz [36]. Recently, Kobayashi [23] designed a polynomial-time algorithm for the weighted C 3 -free 2-matching problem in which the cycles of length three are edge-disjoint.…”