Broadcast Domination Number Research Articles

A note on bounds for the broadcast domination number of graphs

A dominating broadcast of a graph G is a function f:V(G)→{0,1,2,…,diam(G)} such that f(v)⩽e(v) for all v∈V(G), where e(v) is the eccentricity of v, and for every u∈V(G), there exists a vertex v with f(v)⩾1 and d(u,v)⩽f(v). The cost of f is ∑v∈V(G)f(v). The minimum cost over all the dominating broadcasts of G is called the broadcast domination number γb(G) of G. In this paper, we give new upper and lower bounds for γb(G). We show that both the bounds are tight. Also, we improve the upper bound for a subclass of regular graphs. Further, we explore the broadcast domination numbers of generalized Petersen graphs and a subclass of circulant graphs.

Broadcast Domination in Line Graphs of Trees

A dominating broadcast of a graph G is a function f : V ( G ) → { 0 , 1 , 2 , … , diam ( G ) } such that f ( v ) ⩽ e ( v ) for all v ∈ V ( G ) , where e ( v ) is the eccentricity of v , and for every vertex u ∈ V ( G ) , there exists a vertex v with f ( v ) > 0 and d ( u , v ) ⩽ f ( v ) . The cost of f is ∑ v ∈ V ( G ) f ( v ) . The minimum of costs over all the dominating broadcasts of G is called the broadcast domination number γ b ( G ) of G . A graph $G$ is said to be radial if γ b ( G ) = rad ( G ) . In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph L ( G ) of G , in terms of γ b ( G ) , and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees T for which L ( T ) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the i -subdivision graph of K 1 , n and a subclass of caterpillars are radial. Also, we show that γ b ( L ( C ) ) = γ ( L ( C ) ) for any caterpillar C .

Proving exact values for the $2$-limited broadcast domination number on grid graphs

We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

Broadcast domination of lexicographic and modular products of graphs

A dominating broadcast labeling of a graph G is a function such that for all where e(v) is the eccentricity of v, and for every vertex there exists a vertex v with and The cost of f is The minimum of costs over all the dominating broadcast labelings of G is called the broadcast domination number of G. In this paper, we give bounds to the broadcast domination number of lexicographic product G • H of a connected graph G and a graph H, and we show that the bounds are tight by determining the exact values for lexicographic products of some classes of graphs. Also, we give an algorithm which produces a dominating broadcast labeling of G • H. We use the algorithm to find • and • Further, we give an upper bound for the broadcast domination number of the modular product of two connected graphs and exact values are determined for the modular product of two graphs involving path, cycle and complete graphs.

Global dominating broadcast in graphs

A broadcast on a graph [Formula: see text] is a function [Formula: see text] such that for each [Formula: see text], where [Formula: see text] denotes the diameter of [Formula: see text] and [Formula: see text] denotes the eccentricity of [Formula: see text]. The cost of a broadcast is the value [Formula: see text]. In this paper, we define and study a new invariant of broadcasts in graphs, which is global dominating broadcast. A dominating broadcast [Formula: see text] of a graph [Formula: see text] is a global dominating broadcast if [Formula: see text] is also a dominating broadcast of [Formula: see text] the complement of [Formula: see text]. We begin by determining the global broadcast domination number of bipartite graphs, paths, cycles, grid graphs and trees. Then we establish lower and upper bounds on the global broadcast domination number of a graph. Finally, we establish relationships between the global broadcast domination number and other parameters.

Dominating Broadcasts in Fuzzy Graphs

Broadcasting problems in graph theory play a significant role in solving many complicated physical problems. However, in real life there are many vague situations that sometimes cannot be modeled using usual graphs. Consequently, the concept of a fuzzy graph GF:(V,σ,μ) has been introduced to deal with such problems. In this study, we are interested in defining the notion of dominating broadcasts in fuzzy graphs. We also show that, in a connected fuzzy graph containing more than one element in σ*, a dominating broadcast always exists, where σ* is {v∈V|σ(v)>0}. In addition, we investigate the relationship between broadcast domination numbers, radii, and domination numbers in a fuzzy graph as follows; γb(GF)≤min{r(GF),γ(GF)}, where γb(GF) is the broadcast domination number, r(GF) is the radius, and γ(GF) is domination numbers in fuzzy graph GF, with |σ*|>1.

On (<i>t, r</i>) Broadcast Domination of Directed Graphs

A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.

(t,r) Broadcast domination number of the join and corona of graphs

(t,r) Broadcast domination number of the join and corona of graphs

The [formula omitted] broadcast domination number of some regular graphs

The (t,r) broadcast domination is a generalization of domination in a graph. In this paper we prove a conjecture by Drews, Harris, and Randolph about the minimal density of towers in Z2 that provide a (t,3) domination broadcast for t≥17 and explore generalizations. Additionally, we determine the (t,r) broadcast domination number of powers of paths, Pn(k) and powers of cycles, Cn(k).

2-limited broadcast domination in subcubic graphs

In this paper, we consider a limited version of the well-studied dominating broadcast function of a graph. For a graph G, a function f:V(G)→{0,1,2} is a 2-limited dominating broadcast if for each vertex u in G there exist a vertex v such that f(v)>0 and dG(u,v)≤f(v), where dG(u,v) denotes the distance between u and v in G. The cost of a 2-limited dominating broadcast f is the sum of the function values f(v) summed over all vertices v in G. The 2-limited broadcast domination number of G, denoted by γb,2(G), is the minimum cost of a 2-limited dominating broadcast in G. We observe that γb,2(G)≤γ(G), where γ(G) is the domination number of G. A graph is (C4,C6)-free if it does not contain a 4-cycle or a 6-cycle as an induced subgraph. We prove that if G is a cubic graph of order n that is (C4,C6)-free, then γb,2(G)≤13n.

Bounds on the sum of broadcast domination number and strong metric dimension of graphs

Let [Formula: see text] be a connected graph of order at least two with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text]. A function [Formula: see text] is called a dominating broadcast function of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], and the broadcast domination number, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all dominating broadcast functions [Formula: see text] of [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the set of vertices [Formula: see text] such that either [Formula: see text] lies on a [Formula: see text] geodesic or [Formula: see text] lies on a [Formula: see text] geodesic of [Formula: see text]. Let [Formula: see text] be a function and, for any [Formula: see text], let [Formula: see text]. We say that [Formula: see text] is a strong resolving function of [Formula: see text] if [Formula: see text] for every pair of distinct vertices [Formula: see text], and the strong metric dimension, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all strong resolving functions [Formula: see text] of [Formula: see text]. For any connected graph [Formula: see text], we show that [Formula: see text]; we characterize [Formula: see text] satisfying [Formula: see text] equals two and three, respectively, and characterize unicyclic graphs achieving [Formula: see text]. For any tree [Formula: see text] of order at least three, we show that [Formula: see text], and characterize trees achieving equality. Moreover, for a tree [Formula: see text] of order [Formula: see text], we obtain the results that [Formula: see text] if [Formula: see text] is central, and that [Formula: see text] if [Formula: see text] is bicentral; in each case, we characterize trees achieving equality. We conclude this paper with some remarks and an open problem.

Efficient [formula omitted] broadcast dominating sets of the triangular lattice

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph G called the (t,r)broadcast domination number which depends on the positive integer parameters t and r. In this setting, a v∈V is a broadcast vertex of transmission strengtht if it transmits a signal of strength t−d(u,v) to every vertex u∈V with d(u,v)<t. Given a set of broadcast vertices S⊆V, the reception at vertex u is the sum of the transmissions from the broadcast vertices in S. The set S⊆V is called a (t,r)broadcast dominating set if every vertex u∈V has a reception strength r(u)≥r and for a finite graph G the cardinality of a smallest broadcast dominating set is called the (t,r) broadcast domination number of G. In this paper, we consider the infinite triangular grid graph and define efficient(t,r) broadcast dominating sets as those broadcasts that minimize signal waste. Our main result constructs efficient (t,r) broadcasts on the infinite triangular grid graph for all t≥r≥1. Using these broadcasts, we then provide upper bounds for the (t,r) broadcast domination numbers for triangular matchstick graphs when (t,r)∈{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(t,t)}.

On the broadcast domination number of permutation graphs

Broadcast domination models the idea of covering a network of cities by transmitters of varying powers while minimizing the total cost of the transmitters used to achieve full coverage. To be exact, let G be a connected graph of order at least two with vertex set V(G) and edge set E(G). Let d(x,y), e(v), and diam(G), respectively, denote the length of a shortest x−y path in G, the eccentricity of a vertex v in G, and the diameter of G. A function f:V(G)→{0,1,…,diam(G)} is called a broadcast if f(v)≤e(v) for each v∈V(G). A broadcast f is called a dominating broadcast of G if, for each vertex u∈V(G), there exists a vertex v∈V(G) such that f(v)>0 and d(u,v)≤f(v). The broadcast domination number, γb(G), of G is the minimum of ∑v∈V(G)f(v) over all dominating broadcasts f on G. Let G1 and G2 be two disjoint copies of a graph G, and let σ:V(G1)→V(G2) be a bijection. Then a permutation graphGσ=(V,E) has vertex set V=V(G1)∪V(G2) and edge set E=E(G1)∪E(G2)∪{uv:v=σ(u)}. For a connected graph G of order at least two, we prove the sharp bounds 2≤γb(Gσ)≤2γb(G); we give an example showing that there is no function h such that γb(G)<h(γb(Gσ)) for all pairs (G,σ). We characterize Gσ satisfying γb(Gσ)=2, and examine γb(Gσ) when G is a cycle, a path, or a complete multipartite graph.

Broadcast domination and multipacking in strongly chordal graphs

A linear programming algorithm to compute the broadcast domination number and the multipacking number of a strongly chordal graph is described. It runs in time cubic in the number of vertices of the input graph. A new linear-time algorithm to find the broadcast domination number of a tree is given, as is a simplified linear algorithm to find the multipacking number of a tree.

Optimal [formula omitted] broadcasts on the infinite grid

Let G=(V,E) be a graph and t,r be positive integers. The reception strength (or signal) that a vertex v receives from a tower of signal strength t located at vertex T is defined as sig(v,T)=max(t−dist(v,T),0), where dist(v,T) denotes the distance between the vertices v and T. In 2015 Blessing, Insko, Johnson, and Mauretour defined a (t,r)broadcast dominating set, or simply a (t,r)broadcast, on G as a set T⊆V such that the sum of all signals received at each vertex v∈V is at least r. We say that T is optimal if |T| is minimal among all such sets T. The cardinality of an optimal (t,r) broadcast on a finite graph G is called the (t,r) broadcast domination number of G. The concept of (t,r) broadcast domination generalizes the classical problem of domination on graphs. In fact, the (2,1) broadcasts on a graph G are exactly the dominating sets of G.In their paper, Blessing et al. considered (t,r)∈{(2,2),(3,1),(3,2),(3,3)} and gave optimal (t,r) broadcasts on Gm,n, the grid graph of dimension m×n, for small values of m and n. They also provided upper bounds on the optimal (t,r) broadcast numbers for grid graphs of arbitrary dimensions. In this paper, we define the density of a (t,r) broadcast, which allows us to provide optimal (t,r) broadcasts on the infinite grid graph for all t≥2 and r=1,2, and bound the density of the optimal (t,3) broadcast for all t≥2. In addition, we present a Python program to compute upper bounds on the density of a minimal (t,r) broadcast on the infinite grid, and compute these bounds for all 1≤t≤15 and 1≤r≤40. Lastly, we construct a family of counterexamples to the conjecture of Blessing et al. that the optimal (t,r) and (t+1,r+2) broadcasts are identical for all t≥1 and r≥1 on the infinite grid.

Formula omitted]-broadcast domination and [formula omitted]-multipacking

We generalize broadcast domination by requiring that every vertex must hear a broadcast from each of k different vertices. Some basic theory of k-broadcast domination and its dual problem, k-multipacking, is developed. We then focus on 2-broadcast domination and show that the 2-broadcast domination number is at most three times the broadcast domination number, but can differ by any additive amount from twice the broadcast domination number. It is shown that the 2-broadcast domination number of a connected graph with n vertices is at most 4n5.

Broadcast Domination in Tori

A broadcast on a graph G is a function f : V (G) ! f0;1;:::;diam(G)g such that for every vertex v 2 V (G), f(v) e(v), where diam(G) is the diameter of G, and e(v) is the eccentricity of v. In addition, if every vertex hears the broadcast, then the broadcast is a dominating broadcast. The cost of a broadcast f is the value (f) = ∑ v2V (G) f(v). In this paper we determine the minimum cost of a dominating broadcast (also known as the broadcast domination number) for a torus Cm □ Cn.

Dominating Broadcast Labeling in Cartesian Products of Graphs

Let γb(G1□G2) denote the broadcast domination number for the Cartesian product of two graphs G1 and G2. In this paper, we evaluate the value of γb(G1□G2), where Gi is either a path, cycle, star, or complete graph.

LIMITED DOMINATING BROADCAST IN GRAPHS

For a given graph G, a function f : V(G) → {0, 1, …, diamG} such that for every vertex v of G, f(v) ≤ e(v) (where e(v) is the eccentricity of the vertex v and diamG is the diameter of G), is a broadcast on G. A broadcast f is a dominating broadcast if Nf[V+] = V(G), where V+ = {u | f(u) &gt; 0}, Nf[V+] = ⋃u∈V+Nf[u], and Nf[u] = {v | d(u, v) ≤ f(u)}. In this paper, inspired by Dunbar et al. [3], we study limited dominating broadcast in graphs and obtain some properties, bounds, and characterizations for limited broadcast domination number in graphs.

New bounds for the broadcast domination number of a graph

Abstract A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) &gt; 0. The cost of a broadcast f is σ(f) = Σv∈V f(v), and the broadcast number λ b (G) is the minimum cost of a dominating broadcast. A set X ⊆ V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λb(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λb.