A New Characterization of Path Graphs
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Path Graphs are intersection graphs of paths in a tree. Path Graphs are closed under taking induced subgraphs and, answering a question posed by Renz [P.L. Renz, Intersection Representations of Graphs by Arcs, Pacific J. Math., 34 (1970), 501-510], Lévêque, Maffray and Preissmann [B. Lévêque, F. Maffray, and M. Preissmann, Characterizing Path Graphs by Forbidden Induced Subgraphs, J. Graph Theory, 62:4 (2009) 369-384] characterized Path Graphs by a list of forbidden minimal subgraphs. In this paper, building on a result due to Monma and Wei [C.L. Monma, and V.K. Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141-181], we introduce the collection of the attachedness graphs of a graph and characterize Path Graphs in three ways: a structural characterization of each of the attachedness graphs, a characterization by a list of five families of minimal forbidden induced 2-edge colored subgraphs in each of the attachedness graphs and, finally, by a list of three families of minimal forbidden (not necessarily induced) 2-edge colored subgraphs in each of the attachedness graphs.
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