Subdivision into i-packings and S-packing chromatic number of some lattices

An $i$-packing in a graph $G$ is a set of vertices at pairwise distance greater than $i$. For a nondecreasing sequence of integers $S=(s\_{1},s\_{2},\ldots)$, the $S$-packing chromatic number of a graph $G$ is the least integer $k$ such that there exists a coloring of $G$ into $k$ colors where each set of vertices colored $i$, $i=1,\ldots, k$, is an $s\_i$-packing. This paper describes various subdivisions of an $i$-packing into $j$-packings (j\textgreater{}i) for the hexagonal, square and triangular lattices. These results allow us to bound the $S$-packing chromatic number for these graphs, with more precise bounds and exact values for sequences $S=(s\_{i}, i\in\mathbb{N}^{*})$, $s\_{i}=d+ \lfloor (i-1)/n \rfloor$

Labeled Packing of Cycles and Circuits

In 2013, Duch{\^e}ne, Kheddouci, Nowakowski and Tahraoui [4, 9] introduced a labeled version of the graph packing problem. It led to the introduction of a new parameter for graphs, the k-labeled packing number $\lambda$ k. This parameter corresponds to the maximum number of labels we can assign to the vertices of the graph, such that we will be able to create a packing of k copies of the graph, while conserving the labels of the vertices. The authors intensively studied the labeled packing of cycles, and, among other results, they conjectured that for every cycle C n of order n = 2k + x, with k $\ge$ 2 and 1 $\le$ x $\le$ 2k -- 1, the value of $\lambda$ k (C n) was 2 if x was 1 and k was even, and x + 2 otherwise. In this paper, we disprove this conjecture by giving a counter example. We however prove that it gives a valid lower bound, and we give sufficient conditions for the upper bound to hold. We then give some similar results for the labeled packing of circuits

Eternal dominating sets on digraphs and orientations of graphs

We study the eternal dominating number and the m-eternal dominating number on digraphs. We generalize known results on graphs to digraphs. We also consider the problem "oriented (m-)eternal domination", consisting in finding an orientation of a graph that minimizes its eternal dominating number. We prove that computing the oriented eternal dominating number is NP-hard and characterize the graphs for which the oriented m-eternal dominating number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs and different kinds of grids and products of graphs.Comment: 34 page

The b-chromatic number of power graphs

The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithm

[1,2]-Domination in Generalized Petersen Graphs

A vertex subset $S$ of a graph $G=(V,E)$ is a $[1,2]$-dominating set if each vertex of $V\backslash S$ is adjacent to either one or two vertices in $S$. The minimum cardinality of a $[1,2]$-dominating set of $G$, denoted by $\gamma_{[1,2]}(G)$, is called the $[1,2]$-domination number of $G$. In this paper the $[1,2]$-domination and the $[1,2]$-total domination numbers of the generalized Petersen graphs $P(n,2)$ are determined

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithm

Propagation d’événements dans un graphe économique

International audienceThe diffusion models of infections in social networks are intensively studied these last years. The existing studies concern in particular disease and rumor diffusions in social networks or financial risk in banking networks. We propose in this paper to study the diffusion problem of events within social and economic networks. In particular, we define a new problem of diffusion called the Influence Classification Problem. The objective is to find the set of nodes which are impacted by a given network. We also propose two diffusion models based on a computed threshold according to the graph and event attributes. We test our models ontwo real and known events : the hurricane Katrina and the fusion of Bayer and MonsantoLes modèles de diffusion dans les réseaux sociaux sont beaucoup étudiés ces dernières années. Les études concernent notamment les diffusions de maladies et de rumeurs dans les réseaux sociaux ou de risques financiers dans les réseaux bancaires. Nous proposons dans cet article de répondre au problème de diffusion des événements au sein de réseaux économico-sociaux. En particulier, nous proposons d’étudier un nouveau problème de diffusion appelé Influence Classification Problem (ICP) dont l’objectif est de classifier automatiquement quels noeuds sont impactés pour un événement donné. Nous proposons également deux modèles de propagation basés sur un seuil calculé en fonction desattributs du graphe et de l’événement. Nous testons nos modèles sur deux événements connus : l’ouragan Katrina et l’acquisition de Monsanto par Bayer