Abstract
In this work we assess subclasses of chordal graphs focusing on special structures such as maximal cliques and minimal vertex separators. A subclass is presented and characterized, the non-inclusion chordal graphs, graphs in which there is no proper containment between any two minimal vertex separators. A hierarchical structure of classes is outlined, showing that some well-known subclasses of chordal graphs are subclasses of the newly defined class. In order to characterize non-inclusion chordal graphs, a supporting structure, the clique-bipartite graph of a chordal graph is defined.
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