New bounds for the broadcast domination number of a graph

Transcription

1 Cent. Eur. J. Math. 11(7) DOI: /s Central European Journal of Mathematics New bounds for the broadcast domination number of a graph Research Article Richard C. Brewster 1, Christina M. Mynhardt 2, Laura E. Teshima 2 1 Department of Mathematics and Statistics, Thompson Rivers University, 900 McGill Road, Kamloops, BC, V2C 0C8, Canada 2 Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, BC, V8W 3P4, Canada Received 5 May 2012; accepted 31 July 2012 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) > 0. The cost of a broadcast f is σ(f) = v V f(v), and the broadcast number γ 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 γ b (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 γ b. MSC: 05C69, 05C70 Keywords: Broadcast domination Broadcast number Irredundance Multipacking Versita Sp. z o.o. 1. Introduction The main purpose of this paper is to present a new tight upper bound for the broadcast number of a graph in terms of its irredundance number. These two parameters have been associated frequently with domination but never before with each other. rbrewster@tru.ca mynhardt@math.uvic.ca lteshima@uvic.ca 1334

2 R.C. Brewster, Ch.M. Mynhardt, L.E. Teshima Broadcast domination can also be considered as an integer programming (IP) problem, which, as usual, may be relaxed to a linear programming (LP) problem. By considering the dual LP problem and its associated IP problem we obtain a generalization of 2-packings, which we call multipackings. The associated parameter, the multipacking number, provides a lower bound for the broadcast number. A 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. We say that f is a dominating broadcast of G if every vertex of G is within distance f(v) from a vertex v such that f(v) > 0. The cost of a broadcast f is given by σ(f) = v V f(v), and the broadcast number of G is given by γ b (G) = min {σ(f) : f is a dominating broadcast of G}. A dominating broadcast f of G such that σ(f) = γ b (G) is called a γ b -broadcast. For vertices u, v of G, we say that v dominates u if u = v or u is adjacent to v. 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; it is possible that y = x. An irredundant set X is maximal irredundant if, for any v V X, X {v} is not irredundant. The irredundance number ir(g) is the cardinality of a smallest maximal irredundant set of G. A maximal irredundant set of cardinality ir(g) is an ir-set. Having dispensed with these basic definitions we now state our main theorem. The proof is given in Section 4 after the presentation of more definitions and background information in Section 2 and a brief discussion of the relevant IP and LP problems in Section 3. Theorem 1.1. For any graph G, γ b (G) 3ir(G)/2. Moreover, for each k Z + there exists a connected graph G k such that ir(g k ) = 2k and γ b (G) = 3k. 2. Definitions, notation and background We follow the notation of [4, 13]. Consider a broadcast f on the graph G. The broadcast vertices of G form the set V f + = {v V : f(v) > 0}. The broadcast neighbourhood of v V f + is the set N f [v] = {u V : d(u, v) f(v)}. Thus, f is a dominating broadcast of G, or G is f-dominated, if for each u V there exists v V f + such that u N f [v]; we say that u hears the broadcast from v. If f : V {0, 1} is a dominating broadcast, then V f + is a dominating set of G. The domination number γ(g) is the cardinality of a smallest dominating set of G. Erwin [11, 12] was the first to consider the broadcast domination problem, and mentioned the trivial bound γ b (G) min{rad G, γ(g)} for any graph G. Other works on broadcast domination include [7, 9, 10, 16 18, 21]. In addition, Heggernes and Lokshtanov [15] and Dabney, Dean and Hedetniemi [8] showed that minimum broadcast domination is solvable in polynomial time for any graph and in linear time for trees, respectively. On the other hand, Pfaff [20] demonstrated that the problem of determining the irredundance number of arbitrary graphs is NP-hard. Thus the bound in Theorem 1.1 provides a lower bound for the irredundance number in terms of the more easily computed broadcast number. The open and closed neighbourhoods of X V are denoted by N(X) and N[X], respectively, and N({v}), N[{v}] are abbreviated, as usual, to N(v) and N[v]. For any v X, the private neighbourhood pn(v, X) of v with respect to X is the set of all vertices in N[v] that are not contained in the closed neighbourhood of any other vertex in X, i.e., pn(v, X) = N[v] N[X {v}]. The external private neighbourhood of v with respect to X is the set epn(v, X) = pn(v, X) {v}. If v is an isolated vertex of G[X], then v pn(v, X), otherwise pn(v, X) = epn(v, X). Clearly, then, X is irredundant if and only if pn(v, X) = for all v X. Irredundance was introduced by Cockayne, Hedetniemi and Miller [6]. Bollobás and Cockayne [3] showed that ir(g) γ(g) 2 ir(g) 1 for all graphs G. Since γ b (G) γ(g), we already have the bound γ b (G) 2 ir(g) 1. The bound presented in Theorem 1.1 improves this bound. 1335

3 New bounds for the broadcast domination number of a graph For any positive integer k and any vertex v, we define the (closed) k-neighbourhood of v by N k [v] = {u V : d(u, v) k}. Thus, if f is a broadcast on G such that f(v) = k, then N f [v] = N f(v) [v] = N k [v]. The k-neighbourhood of a set X V is the set N k [X] = v X N k[v]. The vertex set of a graph G can be partitioned into four subsets (see Figure 1): X, an irredundant set, Y = x X epn(x, X), the set of external private neighbours of vertices in X, R = V N[X], the set of vertices not dominated by X, C = V (X Y R), the set of vertices in V X that are dominated by two or more vertices in X. C Z X Y R Figure 1. A graph H illustrating the subsets of V (H). Further, let Z be the set of isolated vertices of G[X]. Cockayne, Grobler, Hedetniemi and McRae [5] provide a useful necessary and sufficient condition for an irredundant set to be maximal irredundant. Theorem 2.1 ([5]). An irredundant set X is maximal irredundant if and only if A: for each v N[R] there exists x X such that pn(x, X) N[v]. If A holds, we say that v annihilates x, and we call A the annihilation property. Henceforth we let X denote a maximal irredundant set of G. Then for each r R there exists x X such that d(r, x) = 2, that is, R N 2 [X]. Let R(x) = {r R : d(r, x) = 2}. Note that if x, x X, then R(x) and R(x ) may or may not be disjoint. Moreover, by the annihilation property, for each r R there exists x X such that epn(x, X) N(r). Note that r R(x) in this case. Hence R = x X R(x). (1) 3. Broadcast domination as an LP problem Like many other graph-theoretic parameters, broadcast domination can be considered as an integer programming (IP) problem. Its fractional relaxation linear program (LP) has a dual linear program (DLP) whose IP formulation provides a lower bound for the broadcast number, see (2) below, which we use to establish γ b (G k ) for the graph G k in Theorem 1.1. We refer the reader to [14, Chapter 1] by P.J. Slater for a discussion of LP-duality in domination-related problems. Other papers on broadcast domination algorithms include [1, 2, 8, 15, 20, 22]. 1336

4 R.C. Brewster, Ch.M. Mynhardt, L.E. Teshima A dominating broadcast on a graph G can also be viewed as a covering of G with k-neighbourhoods centred at each of the broadcast vertices. Thus a broadcast can be seen as a collection B = {N k [v]} such that for each u V there exists some N k [v] B with u N k [v]. For this covering form of broadcasting, we denote the cost of the broadcast B by σ B = k. N k [v] B Finding the minimum σ B is a natural IP. With each N k [v] we associate an indicator variable x k,v {0, 1}, where The IP objective function is given by x k,v = min v V { 1 if f(v) = k, 0 otherwise. 1 k e(v) k x k,v. There is one constraint for each vertex u. We define B u = {(k, v) : u N k [v]}, the set of k-neighbourhoods that contain u. Our IP constraints require that each u be in at least one selected k-neighbourhood. That is, for each u V, (k,v) B u x k,v 1. The fractional relaxation LP is given by min k x k,v v V 1 k e(v) s.t. (k,v) B u x k,v 1 for each u V, x k,v 0. The dual LP has one variable y u for each vertex u. It is max y u s.t. y u k u V for each N k [v], y u 0. u N k [v] That is to say, we assign a weight y u to each u V so that, for each k {1,..., e(u)}, the total weight in the k-neighbourhood of u does not exceed k. In the case that y u {0, 1} this simplifies to choosing a set of vertices Y so that each k-neighbourhood of u has at most k vertices in Y. The following proposition is an immediate consequence of these concepts. Proposition 3.1. Let Y V be such that Y N k [v] k for each v V and each k = 1, 2,..., e(v). Then γ b Y. Proof. Let f be a γ b -broadcast and v V +. Then N f [v] contains at most f(v) vertices from Y. Since V = v V + N f[v], we obtain γ b = f(v) N f [v] Y Y. v V + f v V + f Consider a graph G and an integer k such that 1 k diam G. A set Y of vertices of G is called a k-multipacking if, for each v V and each r such that 1 r k, N r [v] contains at most r vertices in Y. The k-multipacking number mp k (G) is the maximum cardinality of a k-multipacking of G. A 1-multipacking is simply an ordinary 2-packing. If Y is a k-multipacking, where k = diam G, we simply call Y a multipacking, while the k-multipacking number of G is then called the multipacking number and denoted mp(g). Hence a set Y that satisfies the conditions of Proposition 3.1 is a multipacking of G, and by Proposition 3.1, γ b (G) mp(g). (2) For some graphs it turns out that γ b (G) = mp(g), that is, the PLP and DLP have optimum integer solutions. However, for other graphs γ b (G) > mp(g). This is the case when the PLP and DLP have fractional optima. Consider the example of C 5, where x 1,v = y v = 1/3 and x 2,v = 0 for each vertex v. Then v V 2 k=1 kx k,v = u V y u = 5/3. Thus these labellings, which satisfy the constraints, are primal and dual optimal, and min v V 2 k=1 kx k,v = max y u = 5/3. However, γ b (C 5 ) = 2 and mp(c 5 ) =

5 New bounds for the broadcast domination number of a graph 4. Proof of Theorem 1.1 To prove our new upper bound, we first prove two lemmas. We use the notation defined in Section 2. Lemma 4.1. For any z Z and any r R(z), there exists x X Z such that r R(x). Proof. By the definition of R, r / N[X]. By the annihilation property of the maximal irredundant set X, r annihilates some x X. Thus r R(x). Since r is not adjacent to x, x / N[r]. Since r annihilates x, pn(x, X) N[r]. It follows that x / pn(x, X), that is, x is not an isolated vertex of G[X]. Therefore x X Z, as required. By Lemma 4.1 and (1), R = x X Z R(x), that is, R N 2 [X Z]. (3) Lemma 4.2. If a graph G has an ir-set X such that G[X] has only isolated vertices, then γ b (G) ir(g). Proof. By Lemma 4.1, R =. Hence X dominates G so that γ b (G) γ(g) = ir(g). We now prove Theorem 1.1, which we restate here for convenience. Theorem 1.1. For any graph G, γ b (G) 3 ir(g)/2. Moreover, for each k Z + there exists a connected graph G k such that ir(g k ) = 2k and γ b (G) = 3k. Proof. Let X be an ir-set of G. We proceed by examining three types of components of G[X] and their 2-neighbourhoods. To simplify notation we denote the vertex set of each component X i of G[X] also by X i. We define a broadcast f such that V f + X and for each X i = K 2, v X i f(v) X i, while v X i f(v) = 3 if X i = K 2. Hence σ(f) 3 X /2. Type 1: X i is a component with order n 3. Let v be a central vertex of X i. Then, for each x X i, d(v, x) rad X i rad P n = (n 1)/2. Define f(v) = n and f(u) = 0 otherwise. Since the distance from v to any other vertex in N 2 [X i ] is at most (n 1)/2 + 2 n (since n 3), N 2 [X i ] is f-dominated. Type 2: X i = K 2. Type 3: X i = {z}. Choose arbitrary v X i, define f(v) = 3 and f(u) = 0 otherwise. Then N 2 [X i ] N f [v]. Then z Z. Define f(z) = 1. Then N[z] = N f [v]. If X = Z, then by Lemma 4.2 we are done. If X Z, then by (3), R N 2 [X Z]. Hence each vertex in R hears a broadcast from a vertex in a Type 1 or 2 component of G[X]. Also, C N[X], hence each vertex in C hears a broadcast from a vertex in X. It follows that f is a dominating broadcast of G and so γ b (G) σ(f) 3 X /2. Let H be the graph depicted in Figure 2. Since H has no universal vertex (a vertex adjacent to every other vertex of H), ir(h) 2. Also, X = {v 3, v 4 } is irredundant and we only need to show maximality. Since pn(v 3, X) = {v 2 } and pn(v 4, X) = {v 5 }, X {y} is redundant for each y N[R] = {v 1, v 1, v 2, v 5, v 6, v 6 }. This shows that ir(h) = 2. The function f : V (H) {0, 3} defined by f(v 3 ) = 3 and f(u) = 0 otherwise is a dominating broadcast of H, hence γ b (H) 3. Suppose H has a dominating broadcast g with cost 2. Let w be the vertex that broadcasts to v 1. Since g(w) 2, w {v 1, v 1, v 2, v 3 }, hence w does not broadcast to v 6. Therefore g(w) = 1, w {v 1, v 1, v 2}, and there exists a vertex w w with g(w ) = 1 that broadcasts to v 6. As for w, w {v 5, v 6, v 6 }. Now v 3 hears no broadcast, a contradiction. Therefore γ b (H) = 3. For i = 1,..., k, let H i = H, and for j = 0,..., 5, label the vertex of Hi corresponding to the vertex v 6 j (v 6 j, where applicable) of H by v 6i j (v 6i j ). Form the graph G k by joining v 6i to v 6i+1, i = 1,..., k 1. See Figure

6 R.C. Brewster, Ch.M. Mynhardt, L.E. Teshima v 3 X 3 v 3 v 4 v 2 v 5 v 1 v 1 v 6 v 6 Figure 1: A graph H with ir(h) = 2 and γ b (H) = 3 Figure 2. A graph H with ir(h) = 2 and γ b (H) = 3. v 3 v 9 v 6k 3 X v 3 v 4 v 9 v 10 v 6k 3 v 6k 2 v 2 v 5 v 8 v 11 v 6k 4 v 6k 1 v 1 v 6 v 7 v 12 v 6k 5 v 6k v 1 v 6 v 7 v 12 v 6k 5 v 6k Figure 3. A graph G k with ir(g k ) = 2k and γ b (G k ) = 3k. 1 As for H, X = k i=1 {v 6i 3, v 6i 2 } is a maximal irredundant set of G k of cardinality 2k, hence ir(g k ) 2k. We show that γ b (G k ) = 3k; it will follow that ir(g k ) 2γ b (G k )/3 = 2k. The function f : V (G k ) {0, 3} defined by f(u) = 3 if u {v 6i 3 : i = 1,..., k} and f(u) = 0 otherwise is a dominating broadcast of G k, hence γ b (G k ) 3k. It is easy to see that the set Y = { v 1, v 3, v 6, v 7, v 9, v 12,..., v 6k 5, v 6k 3, 6k} v, indicated by larger circles in Figure 3, is a multipacking of G k. Hence, by Proposition 3.1, γ b (G k ) mp(g k ) Y = 3k. This completes the proof of Theorem Corollaries In proving the next two corollaries to Theorem 1.1 we often define a broadcast f on a component of an ir-set as for the Type 1, 2 or 3 components in the proof of Theorem 1.1. We then simply say that we use a Type i assignment for f, where i {1, 2, 3}. Corollary 5.1. If a graph G has an ir-set X such that every nontrivial component of G[X] has order at least three, then γ b (G) ir(g). Proof. By using a Type 1 or Type 3 assignment for each component of G[X], we define a dominating broadcast f with σ(f) = X. 1339

7 New bounds for the broadcast domination number of a graph Corollary 5.2. Suppose γ b (G) = 3 ir(g)/2. Then, for any ir-set X of G, (i) G[X] = mk 2 for some integer m, (ii) each vertex c C is adjacent to exactly two vertices u, v X, where uv E(G), and c is adjacent to no vertex in N 2 [x] where x X {u, v}, (iii) for each u X there exists r R(u) such that r / R(w) for any w X {u}, (iv) for each edge uv of G[X] there exists c C adjacent to u and v, (v) for any two vertices u, v X, no vertex in pn(u, X) is adjacent to all vertices in pn(v, X). Proof. Assume γ b (G) = 3 ir(g)/2. (i) Suppose G has an ir-set X such that G[X] mk 2. As in the proof of Theorem 1.1, by using a Type 2 assignment on each K 2 component of G[X] and a Type 1 or 3 assignment otherwise, we obtain a dominating broadcast f of G with σ(f) < 3 ir(g)/2, a contradiction. (ii) Suppose c C is adjacent to u, u X, where uu / E(G). Then there exist distinct vertices v, v X adjacent to u and u, respectively. Define a broadcast f so that f(c) = 4, all other components have Type 2 assignments, and f(w) = 0 otherwise. Then f is a dominating broadcast with σ f (G) < 3 ir(g)/2, a contradiction. We now show that c is not adjacent to any vertex in N 2 [X {u, v}]. Case 1: c is adjacent to some c C. By the above, c is adjacent to some component u v of G[X]. Define a broadcast f so that f(c) = 4; for each component in G[X] other than uv and u v, use Type 2 assignments, and f(w) = 0 otherwise. Case 2: c is adjacent to some y pn[u ], where u v is a component of G[X]. Define a broadcast f so that f(y ) = 4; for each component in G[X] other than uv and u v, use Type 2 assignments, and f(w) = 0 otherwise. Case 3: c is adjacent to some r R(u ), where u v is a component of G[X]. Define a broadcast f so that f(c) = 3 and f(v ) = 2; for each edge in G[X] other than uv and u v, use Type 2 assignments, and f(w) = 0 otherwise. In each case f is a dominating broadcast with σ f (G) < 3 ir(g)/2, a contradiction. (iii) Suppose no such r exists and assume u is matched to v in G[X]. Let f(v) = 2, use Type 2 assignments for all other components of G[X], and let f(w) = 0 for all other vertices of G. Then u, pn(u, X), pn(v, X) and R(v) all hear the broadcast from v. By our assumption, for all r R(u), r R(x) for some other x X. Thus, the Type 2 assignment on the component containing x dominates r. Therefore, G is f-dominated and σ(g) < 3 ir(g)/2. This again contradicts the assumption of γ b (G) = 3 ir(g)/2. (iv) Suppose uv is an edge of G[X] such that no vertex in C is adjacent to u and v. Note that pn(u, X) = epn(u, X) and pn(v, X) = epn(v, X). If x pn(u, X) is not adjacent to a vertex in R, then no vertex in R(u) annihilates u. By the annihilation property each r R(u) annihilates a vertex in X {u}, that is, r R(w) for some w X {u}, contradicting (iii). Thus each vertex in pn(u, X) and, similarly, each vertex in pn(v, X) is adjacent to a vertex in R. Choose any a pn(u, X) and any b pn(v, X), and let f(a) = f(b) = 1. Use Type 2 assignments for all other components of G[X], and let f(w) = 0 otherwise. We see immediately that N 2 [X {u, v}] is f-dominated and only need to verify that N 2 [{u, v}] N 2 [X {u, v}] is f-dominated. Consider z N 2 [{u, v}] N 2 [X {u, v}]. Since no vertex in C is adjacent to u and v, z pn(u, X) pn(v, X) R and thus, as shown above, z N[R]. By the annihilation property and the choice of z, pn(w, X) N[z] for w {u, v} and thus z hears the broadcast from either a or b. Therefore f is a dominating broadcast with σ(f) < 3 ir(g), which once again is a contradiction. (v) Suppose u, v X and there is some a pn(u, X) that is adjacent to all vertices in pn(v, X). If uv is an edge in G[X], define a broadcast f by f(a) = 2; for each component of G[X] other than uv, use Type 2 assignments, and f(w) = 0 otherwise. Then σ f (G) < 3 ir(g)/2. Immediately, N[u], N[v] and R(v) are all subsets of N 2 [a], and hear the broadcast from a. We need only examine R(u). Let r R(u). By the annihilation property, r annihilates w X. If w = u, then r N[a] N 2 [a]. If w = v, then r is adjacent to all vertices in pn(v, X) and hence r N 2 [a]. Otherwise, w X {u, v}, in which case r hears the broadcast from w or its unique neighbour in X. In all cases f is a dominating broadcast. 1340

8 R.C. Brewster, Ch.M. Mynhardt, L.E. Teshima Now consider the case when u and v are not adjacent. By (i), let u, v X be the vertices adjacent to u and v respectively. Define f by f(u) = 3 and f(v ) = 2; for any other edge in G[X], use Type 2 assignments, and f(w) = 0 otherwise. Then N 2 [X {v}] is dominated and we only need to check the vertices in R(v). Consider r R(v). By definition, r is adjacent to some b pn(v, X), which in turn is adjacent to a. Thus d(r, u) 3 and r hears the broadcast from u. Moreover, σ(f) < 3 ir(g)/2. This contradiction completes the proof of the corollary. The converse of Corollary 5.1 is not true. The graph G in Figure 4 satisfies conditions (i) (v) but has γ b (G) = 5 and ir(g) = 4. X 5 Figure 4. A counterexample to the converse of Corollary 5.2. Theorem 1.1 shows that the ratio γ b /ir is bounded above by 3/2. However, neither the ratio ir/γ b nor the difference ir γ b is bounded above. Let T be the tree obtained from the star K 1,m by subdividing each edge exactly once. Then ir(t ) = γ(t ) = m and γ b (T ) = Conclusion We have presented a new tight upper bound on the broadcast number of graph, γ b 3ir/2, and a construction of a graph G k such that ir(g k ) = 2k and γ b (G k ) = 3k for any k Z +. This new bound could prove helpful in determining the irredundance number of some graphs by providing a lower bound in polynomial time. Although we have outlined five characteristics of graphs with γ b = 3ir/2, a complete characterization of these graphs has yet to be found. Furthermore, the characterizations of graphs with γ b = 3ir/2, graphs with γ b = ir, and graphs with γ b ir remain open problems. We explicitly restate the open problems here for future reference. Problem 1. Characterize graphs G with γ b (G) = 3 ir(g)/2. Problem 2. Characterize graphs G with γ b (G) = ir(g). Problem 3. Characterize graphs G with γ b (G) ir(g). Denote the 2-packing number of G by ρ(g). It is well known, see, for example, [14, Section 1.2], that the LP relaxations of the IP formulations of the domination and packing problems are duals of each other. This implies that ρ(g) γ(g) for all graphs G. Similarly, (2) states that mp(g) γ b (G) for all G. Since every multipacking is a 2-packing, we also have mp(g) ρ(g) for all graphs G. Meir and Moon [19] showed that if T is a tree, then ρ(t ) = γ(t ). 1341

9 New bounds for the broadcast domination number of a graph Problem 4. Is it true that mp(t ) = γ b (T ) for any tree T? An affirmative answer to Problem 4 would be a nice generalization of Meir and Moon s result. Problem 5. Study multipackings. Acknowledgements Second author is supported by the Natural Sciences and Engineering Research Council of Canada. References [1] Blair J.R.S., Heggernes P., Horton S., Manne F., Broadcast domination algorithms for interval graphs, series-parallel graphs, and trees, In: Proceedings of the Thirty-Fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congr. Numer., 2004, 169, [2] Blair J.R.S., Horton S.B., Broadcast covers in graphs, In: Proceedings of the Thirty-Sixth Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Congr. Numer., 2005, 173, [3] Bollobás B., Cockayne E.J., Graph theoretic parameters concerning domination, independence and irredundance, J. Graph Theory, 1979, 3(3), [4] Chartrand G., Lesniak L., Graphs & Digraphs, 4th ed., Chapman & Hall/CRC, Boca Raton, 2005 [5] Cockayne E.J., Grobler P.J.P., Hedetniemi S.T., McRae A.A., What makes an irredundant set maximal?, J. Combin. Math. Combin. Comput., 1997, 25, [6] Cockayne E.J., Hedetniemi S.T., Miller D.J., Properties of hereditary hypergraphs and middle graphs, Canad. Math. Bull., 1978, 21(4), [7] Cockayne E.J., Herke S., Mynhardt C.M., Broadcasts and domination in trees, Discrete Math., 2011, 311(13), [8] Dabney J., Dean B.C., Hedetniemi S.T., A linear-time algorithm for broadcast domination in a tree, Networks, 2009, 53(2), [9] Dunbar J.E., Erwin D.J., Haynes T.W., Hedetniemi S.M., Hedetniemi S.T., Broadcasts in graphs, Discrete Appl. Math., 2006, 154(1), [10] Dunbar J., Hedetniemi S.M., Hedetniemi S.T., Broadcasts in trees, 2003, manuscript [11] Erwin D.J., Cost Domination in Graphs, PhD thesis, Western Michigan University, Kalamazoo, 2001 [12] Erwin D.J., Dominating broadcasts in graphs, Bull. Inst. Combin. Appl., 2004, 42, [13] Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals of Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 208, Marcel Dekker, New York, 1998 [14] Haynes T.W., Hedetniemi S.T., Slater P.J. (Eds.), Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 209, Marcel Dekker, New York, 1998 [15] Heggernes P., Lokshtanov D., Optimal broadcast domination in polynomial time, Discrete Math., 2006, 306(24), [16] Herke S.R.A., Dominating Broadcasts in Graphs, MSc thesis, University of Victoria, Victoria, 2007, available at [17] Herke S., Mynhardt C.M., Radial trees, Discrete Math., 2009, 309(20), [18] Lunney S., Trees with Equal Broadcast and Domination Numbers, MSc thesis, University of Victoria, Victoria, 2011, available at [19] Meir A., Moon J.W., Relations between packing and covering numbers of a tree, Pacific J. Math., 1975, 61(1),

10 R.C. Brewster, Ch.M. Mynhardt, L.E. Teshima [20] Pfaff J.S., Algorithmic Complexities of Domination-Related Graph Parameters, PhD thesis, Clemson University, Clemson, 1984 [21] Seager S.M., Dominating broadcasts of caterpillars, Ars Combin., 2008, 88, [22] Shen S., Smith J.C., A decomposition approach for solving a broadcast domination network design problem, Ann. Oper. Res., DOI: /s