Exponential Domination in Subcubic Graphs

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if S is a set of vertices of a graph G, then S is an exponential dominating set of G if ∑ v∈S ( 1 2 )dist(G,S)(u,v)−1 > 1 for every vertex u in V (G) \ S, where dist(G,S)(u, v) is the distance between u ∈ V (G) \ S and v ∈ S in the graph G − (S \ {v}). The exponential domination number γe(G) of G is the minimum order of an exponential dominating set of G. In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If G is a connected subcubic graph of order n(G), then n(G) 6 log2(n(G) + 2) + 4 6 γe(G) 6 1 3 (n(G) + 2). For every > 0, there is some g such that γe(G) 6 n(G) for every cubic graph G of girth at least g. For every 0 < α < 2 3 ln(2) , there are infinitely many cubic graphs G with γe(G) 6 3n(G) ln(n(G))α . If T is a subcubic tree, then γe(T ) > 1 6(n(T ) + 2). For a given subcubic tree, γe(T ) can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs.