Strengthening strongly chordal graphs

Abstract

An [Formula: see text]-chord of a cycle [Formula: see text] is a chord that forms a new cycle with a length-[Formula: see text] subpath of [Formula: see text] when [Formula: see text] is at most half the length of [Formula: see text]. Define a graph to be [Formula: see text]-strongly chordal if, for every [Formula: see text], every cycle long enough to have an [Formula: see text]-chord always has an [Formula: see text]-chord. The [Formula: see text]-strongly chordal and [Formula: see text]-strongly chordal graphs are, respectively, the chordal and strongly chordal graphs. Several characterizations of [Formula: see text]-strongly chordal graphs are proved, along with details of the class of [Formula: see text]-strongly chordal graphs.