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For a graph $G,$ the set $D subseteq V(G)$ is a porous exponential dominating set if $1 le sum_{d in D} left( 2 right)^{1-dist(d,v)}$ for every $v in V(G),$ where $dist(d,v)$ denotes the length of the shortest $dv$ path. The porous exponential dominating number of $G,$ denoted $gamma_e^*(G),$ is the minimum cardinality of a porous exponential dominating set. For any graph $G,$ a technique is derived to determine a lower bound for $gamma_e^*(G).$ 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 $mathcal{K_n},$ the Slant Grid $mathcal{S_n},$ and the $n$-dimensional hypercube $Q_n.$
For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is a function $f : V to {0,1,2,3}$ having the property that if $f (v) = 0$, then vertex $v$ must have at least two neighbors assigned $2$ under $f$ or {at least} one neighbor $u$ with
For a graph $G,$ we consider $D subset V(G)$ to be a porous exponential dominating set if $1le sum_{d in D}$ $left( frac{1}{2} right)^{text{dist}(d,v) -1}$ for every $v in V(G),$ where dist$(d,v)$ denotes the length of the smallest $dv$ path. Similar
A vertex $v$ in a porous exponential dominating set assigns weight $left(tfrac{1}{2}right)^{dist(v,u)}$ to vertex $u$. A porous exponential dominating set of a graph $G$ is a subset of $V(G)$ such that every vertex in $V(G)$ has been assigned a sum w
Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programm
Property A is a form of weak amenability for groups and metric spaces introduced as an approach to the famous Novikov higher signature conjecture, one of the most important unsolved problems in topology. We show that property A can be reduced to a