Dominating and irredundant broadcasts in graphs

Abstract

A broadcast on a nontrivial connected 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 . The cost of f is σWe use properties of minimal dominating broadcasts to define the concept of an irredundant broadcast on G. We determine conditions under which an irredundant broadcast is maximal irredundant. Denoting the minimum costs of dominating and maximal irredundant broadcasts by γ b (G) and ir b (G) respectively, the definitions imply that ir b (G) ≤ γ b (G) for all graphs. We show that γ b (G) ≤ 5 4 ir b (G) for all graphs G. We also briefly consider the upper broadcast number Γ b (G) and upper irredundant broadcast number IR b (G), and illustrate that the ratio IR b /Γ b is unbounded for general graphs. Hedetniemi and Miller [6] introduced the concept of irredundance as precisely the property that makes a dominating set minimal dominating.Ahmadi, Fricke, Schroeder, Hedetniemi and Laskar [1] use a property that makes a dominating broadcast minimal dominating, which was first mentioned in [10], to define broadcast irredundance, which we state here in Section 2.4. The broadcast irredundance number of G is defined asThe definitions imply that ir b (G) ≤ γ b (G) for all graphs G, and as our main result we prove that the ratio γ b / ir b is bounded:After defining our basic concepts in Section 2, we present some properties of irredundant broadcasts in Section 3, the most important of which is a necessary and sufficient condition for an irredundant broadcast to be maximal irredundant (Theorem 7). Theorem 1 is proved in Section 4. We briefly discuss upper broadcast domination and irredundance in Section 5, illustrating that the ratio IR b /Γ b is unbounded for general graphs, and conclude with a list of open problems and conjectures in Section 6. DefinitionsThis section contains more definitions concerning dominating broadcasts, neighbourhoods and boundaries of broadcasting vertices, minimal dominating broadcasts and, finally, irredundant broadcasts. For undefined concepts we refer the reader to [5]. Dominating BroadcastsConsider a broadcast f on a connected graph G = (V, E).

Comparing Upper Broadcast Domination and Boundary Independence Numbers of Graphs

“…The upper broadcast number of G is Γ b (G) = max {σ(f ) : f is a minimal dominating broadcast of G} , and a dominating broadcast f of G such that σ(f ) = Γ b (G) is called a Γ b (G)-broadcast (abbreviated to Γ b -broadcast if the graph G is obvious). Introduced by Erwin [11], the upper broadcast number was also studied by, for example, Ahmadi, Fricke, Schroeder, Hedetniemi and Laskar [1], Bouchemakh and Fergani [6], Bouchouika, Bouchemakh and Sopena [8], Dunbar, Erwin, Haynes, Hedetniemi and Hedetniemi [10], and Mynhardt and Roux [15].…”

Lower Bound and Exact Values for the Boundary Independence Broadcast Number of a Tree

“…The upper broadcast number of G is Γ b (G) = max {σ(f ) : f is a minimal dominating broadcast of G}. First defined by Erwin [11], the upper broadcast number was also studied by, for example, Ahmadi, Fricke, Schroeder, Hedetniemi and Laskar [1], Bouchemakh and Fergani [6], Bouchouika, Bouchemakh and Sopena [8], Dunbar, Erwin, Haynes, Hedetniemi and Hedetniemi [10], and Mynhardt and Roux [15].…”

Broadcasts on paths and cycles

“…They confirm the conjectures given in [1] for γ b (P n ), γ b (C n ) and Γ b (P n ), but disprove all other conjectures. The characterization of minimal dominating broadcasts was first given by Erwin in [21], and then restated in terms of private borders 1 by Mynhardt and Roux in [29].…”

“…Proposition 2.1 (Erwin [21], restated in [29] [19] the following bound on the upper broadcast domination number of graphs.…”

“…It is worth pointing out that the difference IR b (G) − Γ b (G) can be arbitrarily large. Indeed, Mynhardt and Roux [29] constructed a family of graphs {G r } r≥3 , where each G r is obtained by joining two copies of K r+1 by r independent edges, and proved the relation Γ b (G r ) = 3 ≤ IR b (G r ) = r for every r ≥ 3. Nevertheless, we may have IR b (G) = Γ b (G), as we will prove in Subsection 3.1 when G is a path or a cycle.…”