Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching
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We give the solution to the following question of C. D. Godsil: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG+≅G, whereG+ is the bipartite multigraph whose adjacency matrix,B+, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.
AMS subject classification (1980)05 C 05 06 A10
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