Computational complexity of the distance constrained labeling problem for trees

All L(p, q)-labeling of a graph is a labeling of its vertices by nonnegative integers such that the labels of adjacent vertices differ by at least p and the labels of vertices at distance 2 differ by at least q. The span of such a, labeling is the maximum label used. Distance constrained labelings are an important graph theoretical approach to the Frequency Assignment Problem applied in mobile and wireless networks. In this paper we show that determining the minimum span of an L(p, q)-labeling of a tree is NP-hard whenever q is not a, divisor of p. This demonstrates significant difference in computational complexity of this problem for q = 1 and q > 1. In addition. we give a sufficient and necessary condition for the existence of an H(p, q)-labeling of a tree in the case when the metric oil the label space is determined by a strongly vertex transitive graph H. This generalizes the problem of distance constrained labeling in cyclic metric, that was known to be solvable in polynomial time for trees.