On the bi-enhancement of chordal-bipartite probe graphs

Given a class C of graphs, a graph G = ( V , E ) is said to be a C -probe graph if there exists a stable (i.e., independent) set of vertices X ⊆ V and a set F of pairs of vertices of X such that the graph G ′ = ( V , E ∪ F ) is in the class C . Recently, there has been increasing interest and research on a variety of C -probe graph classes, such as interval probe graphs, chordal probe graphs and chain probe graphs. In this paper we focus on chordal-bipartite probe graphs. We prove a structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordal-bipartite probe if and only if a certain “enhanced” graph B ∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval probe graphs in Zhang (1994) [18] and to one on chordal probe graphs in Golumbic and Lipshteyn (2004) [11].