Characterizing Path Graphs and Directed Path Graphs using PR-trees

Steven Chaplick

University of Toronto

Abstract

A new data structure, the PR-tree, is introduced to characterize the sets of path-tree models of path graphs. We further characterize the sets of directed path-tree models of directed path graphs with a slight modification to the PR-tree. The characterizations are proven constructively. A careful examination of the algorithmic approach used in the construction shows that a PR-tree that captures the path-tree models of a given graph with n vertices and m edges can be performed in O(a(n+m) * n * m) (where a(s) is the inverse of Ackermann's function arising from Union-Find). Additionally, the size of any PR-tree which is produced by this approach is linear with respect to input graph.