Triangulating Vertex-Colored Graphs
SIAM Journal on Discrete Mathematics
Clique Graphs of Chordal and Path Graphs
SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Chordal Graphs and Their Clique Graphs
WG '95 Proceedings of the 21st International Workshop on Graph-Theoretic Concepts in Computer Science
On the correspondence between tree representations of chordal and dually chordal graphs
Discrete Applied Mathematics
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Many works related to dually chordal graphs, their cliques and neighborhoods were published by Brandstadt et al. (1998) [1] and Gutierrez (1996) [6]. We will undertake a similar study by considering minimal vertex separators and their properties instead. We find a necessary and sufficient condition for every minimal vertex separator to be contained in the closed neighborhood of a vertex and two major characterizations of dually chordal graphs are proved. The first states that a graph is dually chordal if and only if it possesses a spanning tree such that every minimal vertex separator induces a subtree. The second says that a graph is dually chordal if and only if the family of minimal vertex separators is Helly, its intersection graph is chordal and each of its members induces a connected subgraph. We also found adaptations for them, requiring just O(|E(G)|) minimal vertex separators if they are conveniently chosen. We obtain at the end a proof of a known characterization of the class of hereditary dually chordal graphs that relies on the properties of minimal vertex separators.