Finding a Maximum 2-Matching Excluding Prescribed Cycles in Bipartite Graphs

Abstract

We introduce a new framework of restricted 2-matchings close to Hamilton cycles. For an undirected graph (V,E) and a family U of vertex subsets, a 2-matching F is called U-feasible if, for each setU in U, F contains at most |setU|-1 edges in the subgraph induced by U. Our framework includes C_{<=k}-free 2-matchings, i.e., 2-matchings without cycles of at most k edges, and 2-factors covering prescribed edge cuts, both of which are intensively studied as relaxations of Hamilton cycles. The problem of finding a maximum U-feasible 2-matching is NP-hard. We prove that the problem is tractable when the graph is bipartite and each setU in U induces a Hamilton-laceable graph. This case generalizes the C_{<=4}-free 2-matching problem in bipartite graphs. We establish a min-max theorem, a combinatorial polynomial-time algorithm, and decomposition theorems by extending the theory of C_{<=4}-free 2-matchings. Our result provides the first polynomially solvable case for the maximum C_{<=k}-free 2-matching problem for k >= 5. For instance, in bipartite graphs in which every cycle of length six has at least two chords, our algorithm solves the maximum C_{<=6}-free 2-matching problem in O(n^2 m) time, where n and m are the numbers of vertices and edges, respectively.