Excluded $t$-Factors in Bipartite Graphs: Unified Framework for Nonbipartite Matchings, Restricted 2-Matchings, and Matroids

Takazawa, Kenjiro

doi:10.1137/18m1176737

Cited by 2 publications

(2 citation statements)

References 34 publications

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“…For the C 4 -free 2-matching problem in bipartite graphs, Hartvigsen [15] and Pap [37] designed combinatorial polynomial-time algorithms, Babenko [2] improved the running time, and Takazawa [45] showed a decomposition theorem. Takazawa [46,47] extended these results to more generalized classes of Maximum K-Free t-Matching Problem.…”

Section: Further Related Workmentioning

confidence: 93%

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

Iwamasa

Takazawa

2021

Math. Program.

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The problem of finding a maximum 2-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum t-matching which excludes specified complete t-partite subgraphs, where t is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete t-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most 2t − 1. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum 2-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of Bérczi and Végh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most t + 1. Our algorithms are obtained from exploiting the discrete structure of restricted t-matchings and employing an algorithm for the Boolean edge-CSP.

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Section: Further Related Workmentioning

confidence: 93%

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

Iwamasa

Takazawa

2021

Math. Program.

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“…Frank [21] developed an interesting generalization of this problem. Related work has been done by Pap [39], Makai [36], and Takazawa [44][45][46][47]. See also [31] for additional results.…”

Section: Related Workmentioning

confidence: 99%

Finding triangle‐free 2‐factors in general graphs

Hartvigsen

2024

Journal of Graph Theory

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A 2‐factor in a graph is a subset of edges such that every node of is incident with exactly two edges of . Many results are known concerning 2‐factors including a polynomial‐time algorithm for finding 2‐factors and a characterization of those graphs that have a 2‐factor. The problem of finding a 2‐factor in a graph is a relaxation of the NP‐hard problem of finding a Hamilton cycle. A stronger relaxation is the problem of finding a triangle‐free 2‐factor, that is, a 2‐factor whose edges induce no cycle of length 3. In this paper, we present a polynomial‐time algorithm for the problem of finding a triangle‐free 2‐factor as well as a characterization of the graphs that have such a 2‐factor and related min–max and augmenting path theorems.

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Excluded $t$-Factors in Bipartite Graphs: Unified Framework for Nonbipartite Matchings, Restricted 2-Matchings, and Matroids

Cited by 2 publications

References 34 publications

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

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

Finding triangle‐free 2‐factors in general graphs

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