Abstract
A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. G is a unicycle graph if it owns only one cycle.
In [1] Golumbic, Hirst and Lewenstein observed that for a tree or a graph with only odd cycles the size of its maximum uniquely restricted matching is equal to its matching number. They posed the problem of finding other graphs enjoying this equality.
In this paper we give a partial answer to their question proving that if G is a unicycle bipartite graph, then all maximum matchings of G are uniquely restricted if and only if there is an edge e belonging to the cycle such that no maximum matching of G contains e.
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References
M. C. Golumbic, T. Hirst and M. Lewenstein, Uniquely restricted matchings, Algorithmica (2000).
D. Hershkowitz and H. Schneider, Ranks of zero patterns and sign patterns, Linear and Multilinear Algebra 34 (1993) 3–19.
B. Korte, L. Lovasz and R. Schrader, Greedoids, Springer-Verlag, Berlin, 1991.
V. E. Levit and E. Mandrescu, The structure of a-stable graphs, Discrete Mathematics (2000) (accepted).
V. E. Levit and E. Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Applied Mathematics (2000). (accepted)
V. E. Levit and E. Mandrescu, Bipartite graphs with uniquely restricted maximum matchings and their corresponding greedoids, Los Alamos Archive, prE-print arXiv:math. 00 /0011156, 2000, 12 pp.
G. L. Nemhauser and L. E. Trotter, Jr., Vertex packings: structural properties and algorithms, Mathematical Programming 8 (1975) 232–248.
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© 2001 Springer-Verlag London Limited
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Levit, V.E., Mandrescu, E. (2001). Unicycle Bipartite Graphs with Only Uniquely Restricted Maximum Matchings. In: Calude, C.S., Dinneen, M.J., Sburlan, S. (eds) Combinatorics, Computability and Logic. Discrete Mathematics and Theoretical Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-0717-0_13
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DOI: https://doi.org/10.1007/978-1-4471-0717-0_13
Publisher Name: Springer, London
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