Learn More
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)
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt età la diffusion(More)
—Given an undirected graph G = (V, E) and a weight function w : V → R + , a vertex coloring of G is a partition of V into independent sets, or color classes. The weight of a vertex coloring of G is defined as the sum of the weights of its color classes, where the weight of a color class is the weight of a heaviest vertex belonging to it. In the W(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)
A proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu defined the weighted chromatic number of a vertex-weighted graph G as(More)