L(h,1,1)-labeling of outerplanar graphs

An L(h, 1, 1)-labeling of a graph is an assignment of labels from the set of integers {0, ⋯, λ} to the vertices of the graph such that adjacent vertices are assigned integers of at least distance h ≥ 1 apart and all vertices of distance three or less must be assigned different labels. The aim of the L(h, 1, 1)-labeling problem is to minimize λ, denoted by λh,1,1 and called span of the L(h, 1, 1)-labeling. As outerplanar graphs have bounded treewidth, the L(1, 1, 1)-labeling problem on outerplanar graphs can be exactly solved in O(n3), but the multiplicative factor depends on the maximum degree Δ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L(h, 1, 1)-labeling for outer-planar graphs that is within additive constants of the optimum values. © Springer-Verlag Berlin Heidelberg 2006.