Two New Characterizations of Path Graphs

TL;DR

This paper provides two new characterizations of path graphs, including a polynomial recognition algorithm, by reducing the problem to 2-coloring subproblems and analyzing attachedness graphs.

Contribution

It introduces a novel characterization of path graphs through 2-colorings and attachedness graphs, enabling efficient recognition algorithms.

Findings

First characterization directly leads to polynomial recognition

Identification of minimal forbidden subgraphs in attachedness graphs

Reduction of path graph recognition to 2-coloring problems

Abstract

Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by 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] and we reduce it to some 2-colorings 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.