Abstract
This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph.
 In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n>2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.
Highlights
We only consider finite, simple, and undirected graphs
Some natural convexities in graphs are defined by a set P of paths in G, in a way that a set S of vertices of G is convex if and only if for every path P : v0, v1, . . . , vl ∈ P such that v0 and vl belong to S, all vertices of P belong to S
The monophonic convexity is defined by considering P as the set of all induced paths of G [18, 15]
Summary
Simple, and undirected graphs. For a graph G = (V, E), a graph convexity on V is a collection C of subsets of V such that ∅, V ∈ C and C is closed under intersections. If we let P be the set of all paths of G with three vertices, we have the well-known P3-convexity which will be studied in this paper. This convexity was introduced with the aim of modeling the spread of a disease in a grid [5]. Chen [12] shown that the P3-hull number of a graph is hard to approximate within a ratio O(2log1− n), for any > 0, unless NP ⊆ DTIME(npolylog(n)). The aim of this work is twofold, first to contribute to the knowledge of Kneser graphs; second to obtain new formulas for the hull number within a family of graphs having nice structure.
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