A New Characterization of Path Graphs

TL;DR

This paper introduces a new, simplified characterization of path graphs using attachedness graphs, establishing that path graph recognition is in NP∩CoNP and identifying minimal obstructions.

Contribution

It provides a novel characterization of path graphs via attachedness graphs, refining previous descriptions and enabling obstruction-based recognition.

Findings

Path graph membership is in NP∩CoNP.

Characterization reduces vertex coloring to 2-coloring on attachedness graphs.

Identifies minimal forbidden subgraphs as obstructions.

Abstract

Path graphs are intersection graphs of paths in a tree.~In this paper we give a "6\ good characterization" of path graphs, namely, we prove that path graph membership is in $NP\cap CoNP$ without resorting to existing polynomial time algorithms. The characterization is given in terms of the collection of the \emph{attachedness graphs} of a graph, a novel device to deal with the connected components of a graph after the removal of clique separators. On the one hand, the characterization refines and simplifies the characterization of path graphs due to Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection {G}raphs of {P}aths in a {T}ree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181], which we build on, by reducing a constrained vertex coloring problem defined on the \emph{attachedness graphs} to a vertex 2-coloring problem on the same graphs. On the other hand, the characterization allows us to exhibit two exhaustive lists of obstructions to path graph membership in the form of minimal forbidden induced/partial 2-edge colored subgraphs in each of the \emph{attachedness graphs}.