A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms.

The domination number y ( G ) of a graph G = (V E ) is the minimum cardinality of a subset of Vsuch that every vertex is either in the set or is adjacent to some vertex in the set. We show that if a connected graph G has minimum2degree two and is not one of seven exceptional graphs, then y ( g ) I ~l V l .We also characterize those connected graphs with y ( G ) = FJVI.

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A brace is a connected bipartite graph with a perfect matching and at least six vertices such that for every pair of nonadjacent edges, there is a perfect matching containing the edges. We give a method for generating all braces. ß

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