Distance constrained labelings of trees

Abstract

An H(p,q)-labeling of a graph G is a vertex mapping f:V G →V H such that the distance between f(u) and f(v) (measured in the graph H) is at least p if the vertices u and v are adjacent in G, and the distance is at least q if u and v are at distance two in G. This notion generalizes the notions of L(p,q)- and C(p,q)-labelings of graphs studied as particular graph models of the Frequency Assignment Problem. We study the computational complexity of the problem of deciding the existence of such a labeling when the graphs G and H come from restricted graph classes. In this way we extend known results for linear and cyclic labelings of trees, with a hope that our results would help to open a new angle of view on the still open problem of L(p,q)-labeling of trees for fixed p>q>1 (i.e., when G is a tree and H is a path). We present a polynomial time algorithm for H(p,1)-labeling of trees for arbitrary H. We show that the H(p,q)-labeling problem is NP-complete when the graph G is a star. As the main result we prove NP-completeness for H(p,q)-labeling of trees when H is a symmetric q-caterpillar. © 2008 Springer-Verlag Berlin Heidelberg.