A broadcast domination on a graph assigns an integer value f(u) ≥ 0 to each vertex u, such that every vertex u with f(u) = 0 is within distance f(v) from a vertex v with f(v) > 0. The Broadcast Domination problem seeks to compute a broadcast domination where the sum of the assigned values is minimized. We show that Broadcast Domination can be solved in linear time on block graphs. For general g...

A broadcast on a graph G is a function f : V (G) → {0, 1, . . . , diamG} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V (G). The broadcast number of a graph is the minimum value of ∑ v∈V (G) f(v) among all broadcasts f with the property that each vertex of G is within distance f(v) from a vertex v with f(v) > 0. We characterize a class of trees with equal broadcast and domination n...

a {em roman dominating function} on a graph $g = (v ,e)$ is a function $f : vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. the {em weight} of a roman dominating function is the value $w(f)=sum_{vin v}f(v)$. the roman domination number of a graph $g$, denoted by $gamma_r(g)$, equals the...

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time fo...

a subset $s$ of vertices in a graph $g$ is called a geodetic set if every vertex not in $s$ lies on a shortest path between two vertices from $s$‎. ‎a subset $d$ of vertices in $g$ is called dominating set if every vertex not in $d$ has at least one neighbor in $d$‎. ‎a geodetic dominating set $s$ is both a geodetic and a dominating set‎. ‎the geodetic (domination‎, ‎geodetic domination) number...

In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour 2015. variant, each provides $t-d$ reception to vertex distance $d < t$ from the broadcast. If \ge then no is provided. A considered dominated if it receives $r$ total all broadcasts. Our main results provide some upper lower bounds on density do...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...