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Article overview
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On $(t,r)$ broadcast domination of certain grid graphs | | Natasha Crepeau
; Pamela E. Harris
; Sean Hays
; Marissa Loving
; Joseph Rennie
; Gordon Rojas Kirby
; Alexandro Vasquez
; | | Date: |
16 Aug 2019 | | Abstract: | Let $G=( V(G), E(G) )$ be a connected graph with vertex set $V(G)$ and edge
set $E(G)$. We say a subset $D$ of $V(G)$ dominates $G$ if every vertex in $V
setminus D$ is adjacent to a vertex in $D$. A generalization of this concept
is $(t,r)$ broadcast domination. We designate certain vertices to be towers of
signal strength $t$, which send out signal to neighboring vertices with signal
strength decaying linearly as the signal traverses the edges of the graph. We
let $mathbb{T}$ be the set of all towers, and we define the signal received by
a vertex $vin V(G)$ from a tower $w in mathbb T$ to be $f(v)=sum_{win
mathbb{T}}max(0,t-d(v,w))$. Blessing, Insko, Johnson, Mauretour (2014) defined
a $(t,r)$ broadcast dominating set, or a $(t,r) $ broadcast, on $G$ as a set
$mathbb{T} subseteq V(G) $ such that $f(v)geq r$ for all $vin V(G)$. The
minimal cardinality of a $(t, r)$ broadcast on $G$ is called the $(t, r)$
broadcast domination number of $G$. In this paper, we present our research on
the $(t,r)$ broadcast domination number for certain graphs including paths,
grid graphs, the slant lattice, and the king’s lattice. | | Source: | arXiv, 1908.6189 | | Services: | Forum | Review | PDF | Favorites | |
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