Compactness-Preserving Mapping on Trees

Jan Baumbach, Jiong Guo, Rashid Ibragimov

Abstract: We introduce a generalization of the graph isomorphism problem. Given two graphs G 1 = (V 1, E 1) and G 2 = (V 2, E 2) and two integers l and d, we seek for a one-to-one mapping f:V 1 → V 2, such that for every v ∈ V 1, it holds that L′ v − L v ≤ d, where , , and \(N^i_{G}(v)\) denotes the set of vertices which have distance at most i to v in a graph G. We call this problem Compactness-Preserving Mapping (CPM). In the paper we study CPM with input graphs being trees and present a dichotomy of classical complexity with respect to different values of l and d. CPM on trees can be solved in polynomial time only if l ≤ 2 and d ≤ 1.