In this paper, we explore some properties of identification matrices and exhibit some uses of identification matrices in studying the graph isomorphism problem, a famous open problem. We show that, given two graphs in the form of a certain identification matrix, isomorphism can be tested efficiently in parallel if at least one matrix satisfies the circular 1s property, and more efficiently in parallel if at least one matrix satisfies the consecutive 1s property. Graphs which have identification matrices satisfying the consecutive 1s property include, among others, proper interval graphs and doubly convex bipartite graphs. The result presented here substantially broadens the class of graphs for which there are known efficient parallel isomorphism testing algorithms.
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