Matching preclusion

In graph theory, a branch of mathematics, the matching preclusion number of a graph G (denoted mp(G)) is the minimum number of edges whose deletion results in the elimination of all perfect matchings or near-perfect matchings (matchings that cover all but one vertex in a graph with an odd number of vertices).[1] Matching preclusion measures the robustness of a graph as a communications network topology for distributed algorithms that require each node of the distributed system to be matched with a neighboring partner node.[2]

In many graphs, mp(G) is equal to the minimum degree of any vertex in the graph, because deleting all edges incident to a single vertex prevents that vertex from being matched. This set of edges is called a trivial matching preclusion set.[2] A variant definition, the conditional matching preclusion number, asks for the minimum number of edges the deletion of which results in a graph that has neither a perfect or near-perfect matching nor any isolated vertices.[3][4]

It is NP-complete to test whether the matching preclusion number of a given graph is below a given threshold.[5]

The strong matching preclusion number (or simply, SMP number) is a generalization of the matching preclusion number; the SMP number of a graph G, smp(G) is the minimum number of vertices and/or edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings.[6]

Other numbers defined in a similar way by edge deletion in an undirected graph include the edge connectivity, the minimum number of edges to delete in order to disconnect the graph, and the cyclomatic number, the minimum number of edges to delete in order to eliminate all cycles.

References
  1. Brigham, Robert C.; Harary, Frank; Violin, Elizabeth C.; Yellen, Jay (2005), "Perfect-matching preclusion", Congressus Numerantium, Utilitas Mathematica Publishing, Inc., 174: 185–192.
  2. Cheng, Eddie; Lipták, László (2007), "Matching preclusion for some interconnection networks", Networks, 50 (2): 173–180, doi:10.1002/net.20187.
  3. Cheng, Eddie; Lesniak, Linda; Lipman, Marc J.; Lipták, László (2009), "Conditional matching preclusion sets", Information Sciences, 179 (8): 1092–1101, doi:10.1016/j.ins.2008.10.029.
  4. Park, Jung-Heum; Son, Sang Hyuk (2009), "Conditional matching preclusion for hypercube-like interconnection networks", Theoretical Computer Science, 410 (27–29): 2632–2640, doi:10.1016/j.tcs.2009.02.041.
  5. Dourado, Mitre Costa; Meierling, Dirk; Penso, Lucia D.; Rautenbach, Dieter; Protti, Fabio; de Almeida, Aline Ribeiro (2015), "Robust recoverable perfect matchings", Networks, 66 (3): 210–213, doi:10.1002/net.21624.
  6. Mao, Yaping; Wang, Zhao; Cheng, Eddie; Melekian, Christopher (2018), "Strong matching preclusion number of graphs", Theoretical Computer Science, 713: 11–20, doi:10.1016/j.tcs.2017.12.035.


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