Characterizing s-Strongly Chordal Graphs Using 2-Paths and k-Chords

A well-known 1961 characterization of chordal graphs by G. A. Dirac in terms of simplicial vertices, was extended by Chvátal, Rusu, and Sritharan (Discrete Math., 2002) to a natural sequence of “weakly chordal graphs.” Somewhat similarly, we will start from the same place and characterize a natural sequence of “chordal, strongly chordal, …, s-strongly chordal, …” graphs, except now in terms of 2-path graphs (meaning graphs that are either \(K_3\) or a 2-tree graph with exactly two degree-2 vertices—equivalently, “paths of triangles” that are the 2-connected outerplanar strongly chordal graphs).