On probe permutation graphs

Given a class of graphs G, a graph G is a probe graph of G if its vertices can be partitioned into two sets P, the probes, and N, the nonprobes, where N is an independent set, such that G can be embedded into a graph of G by adding edges between certain vertices of N. If the partition of the vertices into probes and nonprobes is part of the input, then we call the graph a partitioned probe graph of G. In this paper, we provide a recognition algorithm for partitioned probe permutation graphs with time complexity O(n(2)) where n is the number of vertices in the input graph. We show that there are at most O(n(4)) minimal separators for a probe permutation graph. As a consequence, there exist polynomial-time algorithms solving TREEWIDTH and MINIMUM FILL-IN problems for probe permutation graphs.