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J.-C. König, E. Lazard
1994 Discrete Applied Mathematics
A minimum k-broadcast graph is a k-broadcast graph having the minimum number of edges.  ...  In this paper, after giving simple results generalizing previously known results on l-broadcasting, we find minimum k-broadcast graphs for k + 3 < n d 2k + 3.  ...  A minimum k-broadcast graph (mbg) is a k-broadcast graph G on n vertices having B,(n) edges.  ...

Arthur Farley, Stephen Hedetniemi, Sandra Mitchell, Andrzej Proskurowski
1979 Discrete Mathematics
In a minimal broadcast graph G with 2k vertices (k > 0), every vertex must have degree at least k in order to call all 2k vertices in time k.  ...  In order to show that B(2k)e k l 2k-* it suffices to construct a minimal broadcast graph with 2k vertices and k l 2k-' edges.  ...

### Minimum Average Time Broadcast Graphs

Arthur L. Liestman, Nataša Pržulj
1998 Parallel Processing Letters
We initiate the study of minimum average time broadcast graphs -those graphs on n vertices with the fewest edges in which every vertex can broadcast in minimum average time.  ...  We nd minimum average time broadcast graphs for all even n and for more than half of all odd n. In addition, we give some upper and lower bounds on the number of edges in such graphs for all n.  ...  A minimum broadcast graph on n vertices is a broadcast graph with the minimum number of edges over all broadcast graphs on n vertices. This minimum number of edges is denoted B(n).  ...

### Broadcasting in DMA-bound bounded degree graphs

E. Lazard
1992 Discrete Applied Mathematics
Given a connected graph G and a message originator u, the broadcast time of vertex u, b&), is the minimum number of time units required to complete broadcasting from vertex u.  ...  For the complete graph K" with n I 2 vertices, 46,) = r log, + 1 4, yet & is not minimal with respect to this property for any n>% while n> k+ 1.  ...  # + 2) = k+ 1 and a minimum broadcast graph with k + 2 vertices is the star with k+ 1 edges around a centraJ vertex. (3) &-([k + llP) = +kp(k + l)p for pr 1 and a minimum broadcast graph with (k+ l)p vertices  ...

### Lower bounds for the size in four families of minimum broadcast graphs

J.-F. Saclé
1996 Discrete Mathematics
In this paper, we give a lower bound for the size B(n) of a minimum broadcast graph of order n =2 k -4, 2 k -6, 2 k -5 or 2 k -3 which is shown to be accurate in the cases when k =5 and k = 6.  ...  This result provides, together with an upper bound obtained by a construction given in Bermond et al. (1992) , an estimation of the value B(n) for n =2 k -4.  ...  A minimum broadcast graph is a broadcast graph of size B(n). We also call it mbg. It is known  that B(n) --(k -1)n/2, when n =2 k -2, k ~ 4.  ...

### The complexity of broadcasting in bounded-degree networks [article]

Michael J. Dinneen
1994 arXiv   pre-print
We show that for bounded-degree networks determining the minimum broadcast time from an originating node remains NP-complete.  ...  Broadcasting concerns the dissemination of a message originating at one node of a network to all other nodes.  ...  Bounded-Degree Minimum Broadcast Time Problem Instance: Graph G = (V; E) with maximum degree C, vertex v 2 V , and a positive integer k. Question: Is the broadcast time b(v) less than or equal to k?  ...

### On the monotonicity of the broadcast function

Hovhannes A. Harutyunyan, Arthur L. Liestman
2003 Discrete Mathematics
graphs-those graphs which allow k-broadcasting from each vertex in minimum +t time units.  ...  Along the way, we give a method to construct k-broadcast graphs, those graphs which allow minimum time k-broadcasting from each vertex for certain values of n and give some new results on t-relaxed k-broadcast  ...  The k-broadcast function, B k (n), is the minimum number of edges in any k-broadcast graph on n vertices. A minimum k-broadcast graph is a k-broadcast graph on n vertices having B k (n) edges.  ...

Hovhannes A Harutyunyan, Arthur L Liestman
1999 Discrete Applied Mathematics
A minimum broadcast graph on n vertices is a broadcast graph with the minimum number of edges over all broadcast graphs on n vertices. This minimum number of edges is denoted by B(n).  ...  A broadcast graph on n vertices is a graph which allows any vertex to broadcast in time log n .  ...  A minimum broadcast graph on n vertices is a broadcast graph with the minimum number of edges over all broadcast graphs on n vertices. This minimum number of edges is denoted B(n).  ...

Luisa Gargano, Arthur L. Liestman, Joseph G. Peters, Dana Richards
1994 Discrete Applied Mathematics
By adding some redundant calls to the broadcasting scheme, the completion of the broadcast can be guaranteed in the presence of faulty components.  ...  We determine the time required to guarantee a broadcast in this model. We also study the number of links required in networks which allow reliable broadcasting.  ...  A k-reliable minimum broadcast graph (k-rmbg) is a graph G on n vertices with B,(n) edges for which tk(G) = r,(n).  ...

M. Mahéo, J.-F. Saclé
1994 Discrete Applied Mathematics
In this paper, we give the size B(n) of a minimum broadcast graph of order II for the values 20,21 and 22 of n, improve the upper bound of B(23), and by corollary of some other B(n). * Corresponding author  ...  The broadcast function B(n) is the minimum number of edges in any broadcast graph of order n. A minimum broadcast graph is a broadcast graph of size B(n). We also call it mbg.  ...  Fig. 1. 0 3. 4 . 04 Minimum broadcast graph on 22 vertices Theorem 3.7. B(22) = 31Proof. The graph ofFig. 3(a)is a minimum broadcast graph of size 31, thus giving B(22) f 31.  ...

### Upper bounds on the broadcast function using minimum dominating sets

Hovhannes A. Harutyunyan, Arthur L. Liestman
2012 Discrete Mathematics
We construct new upper bounds on the broadcast function B(n), the number of edges in a minimum broadcast graph on n vertices, for a large class of integers n.  ...  The bounds are obtained by a construction that uses the minimum size of dominating sets for some Knödel graphs. We also determine the domatic number of these graphs.  ...  Among all of the broadcast graphs on n vertices, a graph with the fewest edges is called a minimum broadcast graph. This fewest number of edges is the broadcast function, denoted B(n).  ...

### A 1-relaxed minimum broadcast graph on 15 vertices

Tianxing Yao, Guofei Zhou, Jianguo Zhou
2004 Applied Mathematics Letters
In this paper, we give a l-relaxed minimum broadcast graph on 15 vertices which shows that B1(15) = 17, thus reject the conjecture.  ...  In particular, they constructed the sparsest possible time-relaxed broadcast graphs for small n (_< 14) and very sparse time-relaxed broadcast graphs for larger n (< 65).  ...  Therefore, we have k _~ 3. | CLAIM 2. G contains at least one vertex of degree one. Figure 2 . 2 A 1-relaxed minimum broadcast graph on 15 vertices.  ...

### Optimal Broadcast Domination of Arbitrary Graphs in Polynomial Time [chapter]

Pinar Heggernes, Daniel Lokshtanov
2005 Lecture Notes in Computer Science
Broadcast domination assigns a power f (v) ≥ 0 to each vertex v of a given graph, such that every vertex of the graph is within distance f (v) from some vertex v having f (v) ≥ 1.  ...  The optimal broadcast domination problem seeks to minimize the sum of the powers assigned to the vertices of the graph.  ...  Algorithm Minimum Path Broadcast Domination -MPBD Input: A graph G = (V, E).  ...