Two characterizations of chain partitioned probe graphs

Chain graphs are exactly bipartite graphs without induced 2 K 2 (a graph with four vertices and two disjoint edges). A graph G =( V , E ) with a given independent set S ⊆ V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S . In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain “enhanced graph”: G is a chain partitioned probe graph if and only if the enhanced graph G * is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O ( m 2 )-time recognition algorithm for chain partitioned probe graphs.