Elsevier

Discrete Applied Mathematics

Volume 246, 10 September 2018, Pages 49-61
Discrete Applied Mathematics

The S-labeling problem: An algorithmic tour

https://doi.org/10.1016/j.dam.2017.07.036Get rights and content
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Abstract

Given a graph G=(V,E) of order n and maximum degree Δ, the NP-complete S-labeling problem consists in finding a labeling of G, i.e. a bijective mapping ϕ:V{1,2n}, such that SLϕ(G)=uvEmin{ϕ(u),ϕ(v)} is minimized. In this paper, we study the S-labeling problem, with a particular focus on algorithmic issues. We first give intrinsic properties of optimal labelings, which will prove useful for our algorithmic study. We then provide lower bounds on SLϕ(G), together with a generic greedy algorithm, which collectively allow us to approximate the problem in several classes of graphs—in particular, we obtain constant approximation ratios for regular graphs and bounded degree graphs. We also show that deciding whether there exists a labeling ϕ of G such that SLϕ(G)|E|+k is solvable in O(22k(2k)!) time, thus fixed-parameterized tractable in k. We finally show that the S-Labeling problem is polynomial-time solvable for two classes of graphs, namely split graphs and (sets of) caterpillars.

Keywords

Algorithm
Graph labeling

Cited by (0)

A preliminary version of this work appeared in Proceedings of the 26th International Workshop on Combinatorial Algorithms (IWOCA 2015) (Fertin et al., 2015) [10] and in Proceedings of the 5th European Conference on Combinatorics, Graph Theory and Applications(EuroComb 2009) (Fertin and Vialette, 2009) [11].