We consider totally unimodularity for edge-edge adjacency matrices which represent a relationship between two edges in a graph. The matrices appear in integer programming formulations for the minimum maximal matching problem and the edge dominating set problem. We consider a problem of characterizing graphs having totally unimodular matrices as their edge-edge adjacency matrices, and give a necessary and sufficient condition for the characterization. The condition is the first characterization for edge-edge adjacency matrices.
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