2,669 research outputs found

    Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality

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    The Dulmage-Mendelsohn decomposition is a classical canonical decomposition in matching theory applicable for bipartite graphs, and is famous not only for its application in the field of matrix computation, but also for providing a prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn decomposition is stated and proved using the two color classes, and therefore generalizing this decomposition for nonbipartite graphs has been a difficult task. In this paper, we obtain a new canonical decomposition that is a generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs, using a recently introduced tool in matching theory, the basilica decomposition. Our result enables us to understand all known canonical decompositions in a unified way. Furthermore, we apply our result to derive a new theorem regarding barriers. The duality theorem for the maximum matching problem is the celebrated Berge formula, in which dual optimizers are known as barriers. Several results regarding maximal barriers have been derived by known canonical decompositions, however no characterization has been known for general graphs. In this paper, we provide a characterization of the family of maximal barriers in general graphs, in which the known results are developed and unified

    Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm

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    Given an undirected graph and two disjoint vertex pairs s1,t1s_1,t_1 and s2,t2s_2,t_2, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting s1s_1 with t1t_1, and s2s_2 with t2t_2, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions. Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs

    On disjoint matchings in cubic graphs

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    For i=2,3i=2,3 and a cubic graph GG let νi(G)\nu_{i}(G) denote the maximum number of edges that can be covered by ii matchings. We show that ν2(G)≥4/5∣V(G)∣\nu_{2}(G)\geq {4/5}| V(G)| and ν3(G)≥7/6∣V(G)∣\nu_{3}(G)\geq {7/6}| V(G)| . Moreover, it turns out that ν2(G)≤∣V(G)∣+2ν3(G)4\nu_{2}(G)\leq \frac{|V(G)|+2\nu_{3}(G)}{4}.Comment: 41 pages, 8 figures, minor chage
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