Canonical Data Structure for Interval Probe Graphs

Abstract

The class of interval probe graphs is introduced to deal with the physical mapping and sequencing of DNA as a generalization of interval graphs. The polynomial time recognition algorithms for the graph class are known. However, the complexity of the graph isomorphism problem for the class is still unknown. In this paper, extended ${\mathcal MPQ}$-trees are proposed to represent the interval probe graphs. An extended ${\mathcal MPQ}$-tree is canonical and represents all possible permutations of the intervals. The extended ${\mathcal MPQ}$-tree can be constructed from a given interval probe graph in polynomial time. Thus we can solve the graph isomorphism problem for the interval probe graphs in polynomial time. Using the tree, we can determine that any two nonprobes are independent, overlapping, or their relation cannot be determined without an experiment. Therefore, we can heuristically find the best nonprobe that would be probed in the next experiment. Also, we can enumerate all possible affirmative interval graphs for any interval probe graph.