New bounds for the broadcast domination number of a graph

Brewster, Richard C.; Mynhardt, C. M.; Teshima, L. E.

doi:10.2478/s11533-013-0234-8

Cited by 15 publications

(25 citation statements)

References 10 publications

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“…Multipacking The dual notion for Broadcast domination, studied from the linear programming viewpoint, was introduced in [6,30] and called multipacking. A set S of vertices of a (di)graph G is a multipacking if for every vertex v of G and for every possible integer i, there are at most i vertices from S at (directed) distance at most i from v. The multipacking number mp (G) of G is the maximum size of a multipacking in G. Intuitively, if a graph G has a multipacking S, any dominating broadcast of G will require to have cost at least |S| to cover the vertices of S. Hence the multipacking number of G is a lower bound to its broadcast domination number [6]. Equality holds for many graphs, such as strongly chordal graphs [5] and 2-dimensional square grids [1].…”

Section: Broadcast Dominationmentioning

confidence: 99%

On the Complexity of Broadcast Domination and Multipacking in Digraphs

et al. 2021

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We study the complexity of the two dual covering and packing distance-based problems Broadcast domination and multipacking in digraphs. A dominating broadcast of a digraph D is a function f ∶ V(D) → ℕ such that for each vertex v of D, there exists a vertex t with f (t) > 0 having a directed path to v of length at most f(t). The cost of f is the sum of f(v) over all vertices v. A multipacking is a set S of vertices of D such that for each vertex v of D and for every integer d, there are at most d vertices from S within directed distance at most d from v. The maximum size of a multipacking of D is a lower bound to the minimum cost of a dominating broadcast of D. Let Broadcast domination denote the problem of deciding whether a given digraph D has a dominating broadcast of cost at most k, and multipacking the problem of deciding whether D has a multipacking of size at least k. It is known that Broadcast domination is polynomial-time solvable for the class of all undirected graphs (that is, symmetric digraphs), while polynomial-time algorithms for multipacking are known only for a few classes of undirected graphs. We prove that Broadcast domination and multipacking are both NP-complete for digraphs, even for planar layered acyclic digraphs of small maximum degree. Moreover, when parameterized by the solution cost/solution size, we show that the problems are respectively W[2]-hard and W[1]-hard. We also show that Broadcast domination is FPT on acyclic digraphs, and that it does not admit a polynomial kernel for such inputs, unless the polynomial hierarchy collapses to its third level. In addition, we show that both problems are FPT when parameterized by the solution cost/solution size together with the maximum (out-)degree, and as well, by the vertex cover number. Finally, we give for both problems polynomial-time algorithms for some subclasses of acyclic digraphs.

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Section: Broadcast Dominationmentioning

confidence: 99%

On the Complexity of Broadcast Domination and Multipacking in Digraphs

et al. 2021

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“…The upper broadcast domination number is studied in [1,11,19,21,22,29], the broadcast irredundance number is studied in [1,29], and the broadcast independence number is studied in [2,3,4,8,9,13]. Broadcast domination and multipacking are considered in [5,6,15,16,23,30].…”

Section: Dominating Broadcastsmentioning

confidence: 99%

Broadcasts on paths and cycles

Bouchouika

Bouchemakh

Sopena

2020

Discrete Applied Mathematics

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“…The dual notion for Broadcast Domination, studied from the linear programming viewpoint, was introduced in [5,24] and called multipacking. A set S of vertices of a (di)graph G is a multipacking if for every vertex v of G and for every possible integer d, there are at most d vertices from S at (directed) distance at most d from v. The multipacking number mp(G) of G is the maximum size of a multipacking in G. Intuitively, if a graph G has a multipacking S, any dominating broadcast of G will require to have cost at least |S| to cover the vertices of S. Hence the multipacking number of G is a lower bound to its broadcast domination number [5]. Equality holds for many graphs, such as strongly chordal graphs [4].…”

Section: Broadcast Dominationmentioning

confidence: 99%

On the Complexity of Broadcast Domination and Multipacking in Digraphs

Foucaud

Gras

Perez

et al. 2020

Lecture Notes in Computer Science

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We study the complexity of the two dual covering and packing distance-based problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph D is a function f : V (D) → N such that for each vertex v of D, there exists a vertex t with f (t) > 0 having a directed path to v of length at most f (t). The cost of f is the sum of f (v) over all vertices v. A multipacking is a set S of vertices of D such that for each vertex v of D and for every integer d, there are at most d vertices from S within directed distance at most d from v. The maximum size of a multipacking of D is a lower bound to the minimum cost of a dominating broadcast of D. Let Broadcast Domination denote the problem of deciding whether a given digraph D has a dominating broadcast of cost at most k, and Multipacking the problem of deciding whether D has a multipacking of size at least k. It is known that Broadcast Domination is polynomial-time solvable for the class of all undirected graphs (that is, symmetric digraphs), while polynomial-time algorithms for Multipacking are known only for a few classes of undirected graphs. We prove that Broadcast Domination and Multipacking are both NP-complete for digraphs, even for planar layered acyclic digraphs of small maximum degree. Moreover, when parameterized by the solution cost/solution size, we show that the problems are respectively W[2]-hard and W[1]-hard. We also show that Broadcast Domination is FPT on acyclic digraphs, and that it does not admit a polynomial kernel for such inputs, unless the polynomial hierarchy collapses to its third level. In addition, we show that both problems are FPT when parameterized by the solution cost/solution size together with the maximum out-degree. Finally, we give for both problems polynomial-time algorithms for some subclasses of acyclic digraphs.

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New bounds for the broadcast domination number of a graph

Cited by 15 publications

References 10 publications

On the Complexity of Broadcast Domination and Multipacking in Digraphs

On the Complexity of Broadcast Domination and Multipacking in Digraphs

Broadcasts on paths and cycles

On the Complexity of Broadcast Domination and Multipacking in Digraphs

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