Path Graphs, Clique Trees, and Flowers

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}.