Bounding the Porous Exponential Domination Number of Apollonian Networks

Abstract

Given a graph G with vertex set V, a subset S of V is a dominating set if every vertex in V is either in S or adjacent to some vertex in S. The size of a smallest dominating set is called the domination number of G. We study a variant of domination called porous exponential domination in which each vertex v of V is assigned a weight by each vertex s of S that decreases exponentially as the distance between v and s increases. S is a porous exponential dominating set for G if all vertices in S distribute to vertices in G a total weight of at least 1. The porous exponential domination number of G is the size of a smallest porous exponential dominating set. In this paper we compute bounds for the porous exponential domination number of special graphs known as Apollonian networks.