On the L(h, k)‐labeling of co‐comparability graphs and circular‐arc graphs

Abstract

Given two nonnegative integers h and k , an L(h, k )-labeling of a graph G = (V , E ) is a map from V to a set of integer labels such that adjacent vertices receive labels at least h apart, while vertices at distance at most 2 receive labels at least k apart. The goal of the L(h, k )-labeling problem is to produce a legal labeling that minimizes the largest label used. Since the decision version of the L(h, k )-labeling problem is NPcomplete, it is important to investigate classes of graphs for which the problem can be solved efficiently. Along this line of thought, in this article we deal with cocomparability graphs, its subclass of interval graphs, and circular-arc graphs. To the best of our knowledge, ours is the first reported result concerning the L(h, k )-labeling of co-comparability and circular-arc graphs. In particular, we provide the first algorithm to L(h, k )-label co-comparability, interval, and circular-arc graphs with a bounded number of colors. Finally, in the special case where k = 1 and G is an interval graph, our algorithm improves on the best previously-known ones using a number of colors that is at most twice the optimum.

Ordering without Forbidden Patterns

“…A similar situation occurs with BiOrd(F ) with F ⊆ B 4 and classes characterized as BiOrd(F ) with F ⊆ B 4 , including the well known classes of bipartite graphs mentioned earlier. We note that these special graph classes received much attention in the past; efficient recognition algorithms and structural characterizations can be found in [1,3,4,10,16,23,25,27] and elsewhere, cf. [2,13].…”

The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

“…An L(h, k)-labelling algorithm for interval graphs with span at most max(h, 2k)∆ is provided in [33]; this span can be slightly improved under some constraints that the graph has to respect. In the same paper, it is proved that the classical greedy algorithm guarantees a span never larger than min((2h+ 2k − 2)(ω − 1), ∆(2k − 1) + (ω − 1)(2h − 2k)), where ω is the dimension of the larger clique in the graph.…”

“…From the results for interval graphs, the authors of [33] deduce a result on circular arc graphs, i.e. intersection graphs whose model is a set of intervals in a circle.…”

“…A co-comparability graph can be L(h, k)-labeled with span at most max(h, 2k)2∆ + k [33]. This result is obtained exploiting the linear order of the nodes of cocomparability graphs and some considerations on the degree of nodes based on the property of their edges.…”

L(2,1,1)-Labeling of Circular-Arc Graphs

“…An L211L of a graph G = (V, E) is a function γ from its node set V to Z * such that |γ(u) − γ(v)| ≥ 2 if d(u, v) = 1, |γ(u) − γ(v)| ≥ 1 if d(u, v) = 1 or 2. The L211L number, λ 2,1,1 (G), of G is the least non-negative integer λ such that G has a L211L of span λ. Graph labeling problem has been extensively studied in the past [1,7,8,[11][12][13][14][15][16][17][19][20][21][22][23][24]. Different bounds for λ 3,2,1 (G) and λ 4,3,2,1 (G) were obtained for various type of graphs.…”