NÚMERO ENVOLTÓRIO NA CONVEXIDADE P3: RESULTADOS E APLICAÇÕES

Este artigo apresenta uma revisão sistemática da literatura sobre os resultados e aplicações do número envoltório na convexidade P3 em grafos. A determinação deste parâmetro é equivalente ao problema de se encontrar o menor número de vértices de um grafo que permitam disseminar uma informação, influência, ou contaminação, para todos os vértices do grafo. Em particular, esta revisão descreve um panorama sobre estudos teóricos e aplicados acerca do número envoltório P3 considerando a modelagem de fenômenos sociais. Os resultados mostram que o parâmetro é pouco explorado em sociologia computacional para a modelagem de fenômenos sociais. Por outro lado, com o surgimento das redes sociais, pesquisas teóricas têm sido impulsionadas nas últimas décadas. Pesquisadores têm direcionado esforços com o objetivo de contribuir para a solução de problemas relacionados à influência social e disseminação de informação. Entretanto, ainda há espaço para estudos envolvendo o número envoltório na convexidade P3

On interval number in cycle convexity

International audienceRecently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems