Edge-Critical Equimatchable Bipartite Graphs
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A graph is called equimatchable if all of its maximal matchings have the same size. Lesk et al.  provided a characterization of equimatchable bipartite graphs. Since this characterization is not structural, Frendrup et al.  also provided a structural characterization for equimatchable graphs with girth at least five; in particular, a characterization for equimatchable bipartite graphs with girth at least six. In this work, we extend the partial characterization of Frendrup et al.  to equimatchable bipartite graphs without any restriction on girth. For an equimatchable graph, an edge is said to be critical-edge if the graph obtained by removal of this edge is not equimatchable. An equimatchable graph is called edge-critical if every edge is critical. Reducing the characterization of equimatchable bipartite graphs to the characterization of edge-critical equimatchable bipartite graphs, we give two characterizations of edge-critical equimatchable bipartite graphs.
KeywordsEquimatchable Bipartite graphs Edge-critical
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