Graphs with complete minimal k-vertex separators

G. A. Dirac characterized chordal graphs as those in which minimal vertex separators always induce complete subgraphs. I generalize a traditional (2-)vertex separator to a k-vertex separator — meaning a set S of vertices whose removal puts k independent vertices into k separate components. Generalizing Dirac’s theorem, the {P5, 2P3}-free chordal graphs are the graphs in which minimal k-separators always induce complete subgraphs. I discuss other features of k-vertex separators, including their appearance in clique trees of chordal graphs.