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.
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Fiala, J., Golovach, P.A., Kratochvíl, J. (2008). Distance Constrained Labelings of Trees. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_11
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