Domination with exponential decay

Domination with exponential decay

Let G be a graph and S ⊆ V (G). For each vertex u ∈ S and for each v ∈ V (G) − S, we define d(u, v) = d(v, u) to be the length of a shortest path in 〈V (G)−(S−{u})〉 if such a path exists, and∞ otherwise. Let v ∈ V (G). We define wS(v) = ∑ u∈S 1 2d(u,v)−1 if v 6∈ S, and wS(v) = 2 if v ∈ S. If, for each v ∈ V (G), we have wS(v) ≥ 1, then S is an exponential dominating set. The smallest cardinalit...

On exponential domination and graph operations

An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v  in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set....

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 vertice...

On Exponential Domination of Cm × Cn

An exponential dominating set of graph G = (V,E) is a subset D ⊆ V such that ∑ w∈D( 1 2 )d(v,w)−1 ≥ 1 for every v ∈ V, where d(v, w) is the distance between vertices v and w. The exponential domination number, γe(G), is the smallest cardinality of an exponential dominating set. Lower and upper bounds for γe(Cm × Cn) are determined and it is shown that limm,n→∞ γe(Cm×Cn) mn ≤ 1 13 . Two connecti...

Exponential Decay in Nonlinear Networks1