Bounding the Porous Exponential Domination Number of Apollonian Networks
  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. 
Joshua Beverly, Mariah Farley, Christopher McClain, Felicia Stover
math.CO