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A graph is an opposition graph, respectively, a coalition graph, if it admitsan acyclic orientation which puts the two end-edges of every chordless 4-vertexpath in opposition, respectively, in the same direction. Opposition andcoalition graphs have been introduced and investigated in connection to perfectgraphs. Recognizing and characterizing opposition and coalition graphs arelong-standing open problems. This paper gives characterizations for oppositiongraphs and coalition graphs on some restricted graph classes. Implicit in ourarguments are polynomial time recognition algorithms for these graphs. We alsogive a good characterization for the so-called generalized opposition graphs.

Characterization and recognition of some opposition and coalition graph classes