Reduced clique graphs: a correction to "Chordal graphs and their clique graphs"

TL;DR

This paper corrects a proof regarding the characterization of clique trees via reduced clique graphs in chordal graphs and explores structural properties of these graphs, including cycle restrictions and class comparisons.

Contribution

It corrects an error in the proof about maximum weight spanning trees being clique trees and studies structural limitations of reduced clique graphs.

Findings

Corrected the proof of the maximum weight spanning tree characterization.

Proved reduced clique graphs cannot contain induced cycles of length five.

Showed the class of clique graphs is not comparable to reduced clique graphs.

Abstract

Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph $G$. The nodes of the reduced clique graph are the maximal cliques of $G$, and two nodes are joined by an edge if and only if they form a non-disjoint separating pair of cliques in $G$. In this case the weight of the edge is the size of the intersection of the two cliques. A clique tree of $G$ is a tree with the maximal cliques of $G$ as its nodes, where for any $v\in V(G)$, the subgraph induced by the nodes containing $v$ is connected. Galinier et al.\ prove that a spanning tree of the reduced clique graph is a clique tree if and only if it has maximum weight, but their proof contains an error. We explain and correct this error. In addition, we initiate a study of the structure of reduced clique graphs by proving that they cannot contain any induced cycle of length five (although they may contain induced cycles of length three or any even integer greater than two). We show that no cycle of length four or more is isomorphic to a reduced clique graph. We prove that the class of clique graphs of chordal graphs is not comparable to the class of reduced clique graphs of chordal graphs by providing examples that are in each of these classes without being in the other.