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A good edge-labelling of a graph G is a labelling of its edges such that for any two distinct vertices u, v, there is at most one (u, v)-path with non-decreasing labels. This notion was introduced in [3] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of graphs that admit a good(More)
Given a graph G = (V, E), the closed interval of a pair of vertices u, v ∈ V , denoted by I[u, v], is the set of vertices that belongs to some shortest (u, v)-path. The convex hull I h [S] of a subset S ⊆ V is the smallest convex set that contains S. We say that S is a hull set if I h [S] = V. The cardinality of a minimum hull set of G is the hull number of(More)
In this paper, we study a colouring problem motivated by a practical frequency assignment problem and up to our best knowledge new. In wireless networks, a node interferes with the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph G(More)
In this paper, we study the (geodesic) hull number of graphs. For any two vertices u, v ∈ V of a connected undirected graph G = (V, E), the closed interval I[u, v] of u and v is the set of vertices that belong to some shortest (u, v)-path. For any S ⊆ V , let I[S] = u,v∈S I[u, v]. A subset S ⊆ V is (geodesically) convex if I[S] = S. Given a subset S ⊆ V ,(More)
The Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P 4-free graph and N P-hard if G is a P 5-free graph. In this article, we define a new class of graphs, the fat-extended P 4-laden graphs, and we show a polynomial(More)