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

[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

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

Real-time tracking and mining of users’ actions over social media

© 2020, ComSIS Consortium. All rights reserved. With the advent of Web 2.0 technologies and social media, companies are actively looking for ways to know and understand what users think and say about their products and services. Indeed, it has become the practice that users go online using social media like Facebook to raise concerns, make comments, and share recommendations. All these actions can be tracked in real-time and then mined using advanced techniques like data analytics and sentiment analysis. This paper discusses such tracking and mining through a system called Social Miner that allows companies to make decisions about what, when, and how to respond to users’ actions over social media. Questions that Social Miner allows to answer include what actions were frequently executed and why certain actions were executed more than others

A survey on tree matching and XML retrieval

International audienceWith the increasing number of available XML documents, numerous approaches for retrieval have been proposed in the literature. They usually use the tree representation of documents and queries to process them, whether in an implicit or explicit way. Although retrieving XML documents can be considered as a tree matching problem between the query tree and the document trees, only a few approaches take advantage of the algorithms and methods proposed by the graph theory. In this paper, we aim at studying the theoretical approaches proposed in the literature for tree matching and at seeing how these approaches have been adapted to XML querying and retrieval, from both an exact and an approximate matching perspective. This study will allow us to highlight theoretical aspects of graph theory that have not been yet explored in XML retrieval

Remarks on partially square graphs, hamiltonicity and circumference

Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_G(x) ⊆ N_G[u] ∪ N_G[v]$, where $N_G[x] = N_G(x) ∪ {x}$. In the case where G is a claw-free graph, G* is equal to G². We define $σ°ₜ = min{ ∑_{x∈S} d_G(x):S is an independent set in G* and |S| = t}$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results