Convex Partitions of Graphs induced by Paths of Order Three

In [4] Bollobás considers “The spread of an infection on a square grid”. Initially some squares of a square grid are infected. The infection spreads to an uninfected square if at least two of the neighbouring squares are infected. A natural combinatorial problem — solved using an elegant perimeter argument in [4] — is to determine the minimum number of initially infected squares such that eventually all squares become infected. This kind of spreading process has received considerable attention in a variety of contexts such as social influence [15], percolation [3], marketing strategies [10], and distributed computing [25]. In the present paper we study the above process under the perspective of graph convexity. All graphs will be finite, simple, and undirected and we use common terminology and notation. Given a graph G and a collection C of subsets of the vertex set V (G) of G, the pair (G, C) is a graph convexity if ∅ ∈ C, V (G) ∈ C, and C is closed under intersection. The sets in C are called convex. The convex hull of some set S of vertices of some graphGwith respect to some graph convexity (G, C) is the unique smallest set HC(S) in C containing S. Several natural graph convexities are defined in terms of paths. If P is a set of paths in some graph G and C is the collection of all subsets C of V (G) such that, for every path P in P between two vertices in C, the set C contains all vertices of P , then (G, C) is a graph convexity.