Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull-number of a graph and show that computing it is $\Sigma^P_2$ complete.Comment: 13 pages, 3 figure

Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

Strong geodetic problem on Cartesian products of graphs

The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest path between a pair of vertices from $S$. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for ${\rm sg}(G \,\square\, H)$ is determined, as well as exact values for $K_m \,\square\, K_n$, $K_{1, k} \,\square\, P_l$, and certain prisms. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.Comment: 18 pages, 9 figure

On the P3P_3-hull number and infecting times of generalized Petersen graphs

The $P_3$-hull number of a graph is the minimum cardinality of an infecting set of vertices that will eventually infect the entire graph under the rule that uninfected nodes become infected if two or more neighbors are infected. In this paper, we study the $P_3$-hull number for generalized Petersen graphs and a number of closely related graphs that arise from surgery or more generalized permutations. In addition, the number of components of the complement of an infecting set of minimum cardinality is calculated for the generalized Petersen graph and shown to always be $1$ or $2$. Moreover, infecting times for infecting sets of minimum cardinality are studied. Bounds are provided and complete information is given in special cases.Comment: 8 page

Formulas in connection with parameters related to convexity of paths on three vertices: caterpillars and unit interval graphs

We present formulas to compute the P3 -interval number, the P3 -hull number and the percolation time for a caterpillar, in terms of certain sequences associated with it. In addition, we find a connection between the percolation time of a unit interval graph and a parameter involving the diameter of a unit interval graph related to it. Finally, we present a hereditary graph class, defined by forbidden induced subgraphs, such that its percolation time is equal to one.Fil: González, Lucía M.. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Safe, Martin Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin

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