Reduced clique graphs: a correction to "Chordal graphs and their clique graphs"
Dillon Mayhew, Andrew Probert

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 . The nodes of the reduced clique graph are the maximal cliques of , and two nodes are joined by an edge if and only if they form a non-disjoint separating pair of cliques in . In this case the weight of the edge is the size of the intersection of the two cliques. A clique tree of is a tree with the maximal cliques of as its nodes, where for any , the subgraph induced by the nodes containing 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…
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Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research · Interconnection Networks and Systems
