Given a family @math of graphs and a positive integer @math , a graph @math is called vertex @math -fault-tolerant with respect to @math , denoted by @math -FT @math , if @math contains some @math as a subgraph, for every @math with @math . Vertex-fault-tolerance has been introduced by Hayes [A graph model for fault-tolerant computing systems, IEEE Transactions on Computers, C-25 (1976), pp. 875-884.], and has been studied in view of potential applications in the design of interconnection networks operating correctly in the presence of faults. We define the Fault-Tolerant Complete Matching (FTCM) Problem in bipartite graphs of order @math : to design a bipartite @math , with @math , @math , @math , that has a FTCM, and the tuple @math , where @math and @math are the maximum degree in @math and @math , respectively, is lexicographically minimum. @math has a FTCM if deleting at most @math vertices from @math creates @math that has a complete matching, i.e., a matching of size @math . We show that if @math is integer, solutions of the FTCM Problem can be found among @math -regular bipartite graphs of order @math , with @math , and @math . If @math then all @math -regular bipartite graphs of order @math have a FTCM, and for @math , it is not the case. We characterize the values of @math , @math , @math , and @math that admit an @math -regular bipartite graph of order @math , with @math , and give a simple construction that creates such a graph with a FTCM whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.
A problem that is closely related to ours was presented by Perarnau and Petridis @cite_13 . The authors studied the existence of perfect matchings in induced balanced subgraphs of random biregular bipartite graphs.
@cite_13
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We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdos and Renyi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory. A cycle transversal (or feedback vertex set) of a graph G is a subset T@?V(G) such that T@?V(C) 0@? for every cycle C of G. This work considers the problem of finding special cycle transversals in perfect graphs and cographs. We prove that finding a minimum cycle transversal T in a perfect graph G is NP-hard, even for bipartite graphs with maximum degree four. Since G-T is acyclic, this result implies that finding a maximum acyclic induced subgraph of a perfect graph is also NP-hard. Other special properties of T are considered. A clique (stable, respectively) cycle transversal, or simply cct (sct, respectively) is a cycle transversal which is a clique (stable set, respectively). Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known, even for perfect graphs. We characterize cographs with cct in terms of forbidden induced subgraphs and describe their structure. This leads to linear time recognition of cographs with cct. We also prove that deciding whether a perfect graph admits an sct is NP-complete. We characterize cographs with sct in terms of forbidden induced subgraphs; this characterization also leads to linear time recognition. Friedland's Lower Matching Conjecture asserts that if @math is a @math --regular bipartite graph on @math vertices, and @math denotes the number of matchings of size @math , then @math where @math . When @math , this conjecture reduces to a theorem of Schrijver which says that a @math --regular bipartite graph on @math vertices has at least @math perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that @math In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra @math factor compared to the conjecture if @math is separated away from @math and @math , and is tight up to a constant factor if @math is separated away from @math . We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to @math --biregular bipartite graphs. In this note a greedy algorithm is considered that computes a matching for a graph with a given ordering of its vertices, and those graphs are studied for which a vertex ordering exists such that the greedy algorithm always yields maximum cardinality matchings for each induced subgraph. We show that these graphs, called greedy matchable graphs, are a subclass of weakly triangulated graphs and contain strongly chordal graphs and chordal bipartite graphs as proper subclasses. The question when can this ordering be produced efficiently is discussed too.