On the S-labeling Problem

Abstract : Let G be a graph of order n and size m. A labeling of G is a bijective mapping theta : V(G) --> 1, 2...n, and we call Theta(G) the set of all labelings of G. For any graph G and any labeling theta in Theta(G), let SL(G,theta) = sum_{ e in E(G)} min(theta(u) : u \in e). In this paper, we consider the S-Labeling problem, defined as follows: Given a graph G, and a labeling (G) that minimizes SL(G,Theta). The S-Labeling problem has been shown to be NP-complete [Via06]. We prove here basic properties of any optimal S-labeling of a graph G, and relate it to the Vertex Cover problem. Then, we derive bounds for SL(G,Theta), and we give approximation ratios for different families of graphs. We nally show that the S-Labeling problem is polynomial-time solvable for split graphs.
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Guillaume Fertin, Stéphane Vialette. On the S-labeling Problem. Proc. 5th Euroconference on Combinatorics, Graph Theory and Applications (EUROCOMB 2009), 2009, Bordeaux, France. pp.273-277. ⟨hal-00416570⟩

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