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Much of chordal graph theory and its applications is based on chordal graphs being the intersection graphs of subtrees of trees. This suggests also looking at intersection graphs of subgraphs of chordal graphs, and so on, with appropriate conditions imposed on the subgraphs.
This paper investigates such a hierarchy of generalizations of “chordal-type” graphs, emphasizing the so-called “ekachordal graphs” — those next in line beyond chordal
graphs. Parts of the theory of chordal graphs do carry over to chordal-type graphs, including a recursive, elimination characterization for ekachordal graphs.