Path Graphs, Clique Trees, and Flowers

TL;DR

This paper investigates the limitations of asteroidal triples in characterizing path graphs and introduces a new forbidden subgraph family called sun systems for their characterization.

Contribution

It provides a new characterization of path graphs using forbidden sun systems, extending previous work on asteroidal triples and directed path graphs.

Findings

Asteroidal triples are insufficient to characterize path graphs.

Sun systems serve as a new forbidden subgraph family for path graphs.

The characterization generalizes previous results on directed path graphs.

Abstract

An \emph{asteroidal triple} is a set of three independent vertices in a graph such that any two vertices in the set are connected by a path which avoids the neighbourhood of the third. A classical result by Lekkerkerker and Boland \cite{6} showed that interval graphs are precisely the chordal graphs that do not have asteroidal triples. Interval graphs are chordal, as are the \emph{directed path graphs} and the \emph{path graphs}. Similar to Lekkerkerker and Boland, Cameron, Ho\'{a}ng, and L\'{e}v\^{e}que \cite{4} gave a characterization of directed path graphs by a "special type" of asteroidal triple, and asked whether or not there was such a characterization for path graphs. We give strong evidence that asteroidal triples alone are insufficient to characterize the family of path graphs, and give a new characterization of path graphs via a forbidden induced subgraph family that we call \emph{sun systems}. Key to our new characterization is the study of \emph{asteroidal sets} in sun systems, which are a natural generalization of asteroidal triples. Our characterization of path graphs by forbidding sun systems also generalizes a characterization of directed path graphs by forbidding odd suns that was given by Chaplick et al.~\cite{9}.