Dual-Chordal and Strongly Dual-Chordal Graphs

A graph is chordal if and only if every cycle either has a chord or is a triangle. If an edge (or triangle) is defined to be a strength-\(k\) edge (or triangle) whenever it is in at least \( k \) maximum cliques, then a graph is strongly chordal if and only if, for every \( k \geq 1 \), every cycle of strength-\(k\) edges either has a strength-\(k\) chord or is a strength-\(k\) triangle. Dual-chordal graphs have been defined so as to be the natural cycle/cutset duals of chordal graphs. A carefully crafted notion of dual strength allows a characterization of strongly dual-chordal graphs that is parallel to the above. This leads to a complete list of all \( 3 \)-connected strongly dual-chordal graphs.