New graph classes characterized by weak vertex separators and two-pairs

A set of vertices whose deletion from a graph would increase the distance between two remaining vertices is called a weak vertex separator of the graph. Two vertices form a two-pair if all chordless paths between them have length 2. I prove that every inclusion-minimal weak vertex separator of every induced subgraph of a graph G induces a complete subgraph if and only if nonadjacent vertices of induced C4 subgraphs of G always form two-pairs of G; moreover, this is also true when “complete” and C4 are replaced with “edgeless” and K4−e (≅K1,1,2). The first of the resulting new graph classes generalizes chordal graphs, and the second generalizes unichord-free graphs; they both generalize distance-hereditary graphs and geodetic graphs.