Labeling bipartite permutation graphs with a condition at distance two
Discrete Applied Mathematics
| Hi-index | 0.00 |
Given a vector (δ1,δ2,…,δt) of nonincreasing positive integers, and an undirected graph G = (V,E), an L(δ1,δ2,…,δt)-coloring of G is a function f from the vertex set V to a set of nonnegative integers such that ∣f(u) - f(v)∣ ≥ δi, if d(u,v) = i, 1 ≤ i ≤ t, where d(u,v) is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal L(δ1,δ2,…,δt)-coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(δ1,δ2,…,δt)-coloring of two relevant classes of graphs—trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t + δ1)) and O(nt2δ1) time algorithms are proposed to find α-approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and α is a constant depending on t and δ1,…,δt. Moreover, an O(n) time algorithm is given for the L(δ1,δ2)-coloring of unit interval graphs, which provides a 3-approximation. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 204–216 2007