A linear programming method for exponential domination

Abstract

For a graph G, the set D ⊆V (G) is a porous exponential dominating set if 1 ≤ ∑ d∈D (2) 1−dist(d ,v) for every v ∈ V (G), where dist(d , v) denotes the length of the shortest d v path. The porous exponential dominating number of G, denoted γe (G), is the minimum cardinality of a porous exponential dominating set. For any graph G, a technique is derived to determine a lower bound for γe (G). Specifically for a grid graph H , linear programing is used to sharpen bound found through the lower bound technique. Lower and upper bounds are determined for the porous exponential domination number of the King Grid Kn , the Slant Grid Sn , and the n-dimensional hypercube Qn . AMS 2010 Subject Classification: Primary 05C69; Secondary 90C05