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Galinier, Habib, and Paul introduced the reduced clique graph of a chordalgraph 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-disjointseparating pair of cliques in G. In this case the weight of the edge is thesize of the intersection of the two cliques. A clique tree of G is a tree withthe maximal cliques of G as its nodes, where for any v in V(G), the subgraphinduced by the nodes containing v is connected. Galinier et al. prove that aspanning tree of the reduced clique graph is a clique tree if and only if ithas maximum weight, but their proof contains an error. We explain and correctthis error.  In addition, we initiate a study of the structure of reduced clique graphs byproving that they cannot contain any induced cycle of length five (althoughthey may contain induced cycles of length three, four, or six). We show that nocycle of length four or more is isomorphic to a reduced clique graph. We provethat the class of clique graphs of chordal graphs is not comparable to theclass of reduced clique graphs of chordal graphs by providing examples that arein each of these classes without being in the other.

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