A set C of vertices of a graph G is P3-convex if v ∈ C for every path uvw in G with u, w ∈ C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P3-convex sets. Furthermore, we study such partitions for a variety of graph classes.
We study the graphs G for which the hull number h(G) and the geodetic number g(G) with respect to P3-convexity coincide. These two parameters correspond to the minimum cardinality of a set U of vertices of G such that the simple expansion process that iteratively adds to U , all vertices outside of U that have two neighbors in U , produces the whole vertex… (More)
Dynamic monopolies in graphs have been studied as a model for spreading processes within networks. Together with their dual notion, the generalized degenerate sets, they form the immediate generalization of the classical notions of vertex covers and independent sets in a graph. We present results concerning dynamic monopolies in graphs of given average… (More)