Neighborhood-Preserving Mapping between Trees

Jan Baumbach, Jiong Guo, Rashid Ibragimov

Abstract: We introduce a variation of the graph isomorphism problem, where, given two graphs G 1 = (V 1,E 1) and G 2 = (V 2,E 2) and three integers l, d, and k, we seek for a set D ⊆ V 1 and a one-to-one mapping f:V 1 → V 2 such that |D| ≤ k and for every vertex v ∈ V 1 ∖ D and every vertex \(u\in N_{G_1}^l(v)\setminus D\) we have \(f(u)\in N_{G_2}^d(f(v))\). Here, for a graph G and a vertex v, we use \(N_{G}^i(v)\) to denote the set of vertices which have distance at most i to v in G. We call this problem Neighborhood-Preserving Mapping (NPM). The main result of this paper is a complete dichotomy of the classical complexity of NPM on trees with respect to different values of l,d,k. Additionally, we present two dynamic programming algorithms for the case that one of the input trees is a path.