Maximally matched and super matched regular graphs

ABSTRACT In interconnection networks, matching preclusion is a measure of robustness when there is a link failure. Let G be a graph with an even number of vertices. A set of edges of G is called a matching preclusion set if its deletion from G results in a subgraph that has no perfect matchings, and the one with smallest size is called an optimal matching preclusion set, whose cardinality is called matching preclusion number of G, denoted by . G is maximally matched if is equal to the minimum degree of G and is super matched if every optimal matching preclusion set of G consists of edges incident to a single vertex. In this paper, we present a 0–1 integer linear programming for matching preclusion number of general graph. By using perfect matching polytope we obtain simple characterizations for maximally matched and super matched regular graphs. As their applications, we can derive some known results and show some Cartesian product of regular graphs, such as Hamming graph, hyperstar and star cube, are super matched.

[1]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[2]  Dirk Oliver Theis,et al.  Odd minimum cut sets and b-matchings revisited , 2006, SIAM J. Discret. Math..

[3]  Huiqing Liu,et al.  The (conditional) matching preclusion for burnt pancake graphs , 2013, Discret. Appl. Math..

[4]  Eddie Cheng,et al.  Matching preclusion and conditional matching preclusion for bipartite interconnection networks I: Sufficient conditions , 2012, Networks.

[5]  Sajal K. Das,et al.  Folded Petersen cube networks: new competitors for the hypercubes , 1993, Proceedings of 1993 5th IEEE Symposium on Parallel and Distributed Processing.

[6]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[7]  Eddie Cheng,et al.  Matching preclusion and Conditional Matching preclusion for Augmented Cubes , 2010, J. Interconnect. Networks.

[8]  Romeo Rizzi.  A simple minimum T-cut algorithm , 2003, Discret. Appl. Math..

[9]  Sheldon B. Akers,et al.  The Star Graph: An Attractive Alternative to the n-Cube , 1994, ICPP.

[10]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[11]  Eddie Cheng,et al.  Matching preclusion and conditional matching preclusion problems for tori and related Cartesian products , 2012, Discret. Appl. Math..

[12]  Eddie Cheng,et al.  Matching preclusion and conditional matching preclusion for regular interconnection networks , 2012, Discret. Appl. Math..

[13]  Eddie Cheng,et al.  Matching preclusion for some interconnection networks , 2007, Networks.

[14]  Eddie Cheng,et al.  Matching preclusion and conditional matching preclusion problems for the folded Petersen cube , 2015, Theor. Comput. Sci..

[15]  P. D. Seymour,et al.  On Multi‐Colourings of Cubic Graphs, and Conjectures of Fulkerson and Tutte , 1979 .

[16]  Xianyue Li,et al.  Matching preclusion for balanced hypercubes , 2012, Theor. Comput. Sci..

[17]  P. Hall.  On Representatives of Subsets , 1935 .

[18]  Abdel Elah Al-Ayyoub,et al.  The Cross Product of Interconnection Networks , 1997, IEEE Trans. Parallel Distributed Syst..

[19]  Jing Li,et al.  Matching preclusion for k-ary n-cubes , 2010, Discret. Appl. Math..

[20]  Khaled Day,et al.  The Hyperstar Interconnection Network , 1998, J. Parallel Distributed Comput..