Two new characterizations of path graphs

Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei (1986) [14] and we reduce it to some 2-coloring subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.