Limited broadcast domination: upper bounds and complexity  J. C´ aceres a , C. Hernando b , M. Mora b , I.M. Pelayo b , M.L. Puertas a (a) Department of Mathematics, University of Almer´ ıa. E-mail: {jcaceres,mpuertas}@ual.es (b) Department of Mathematics, Polytechnical University of Catalunya E-mail: {carmen.hernando,merce.mora,ignacio.m.pelayo}@upc.edu Abstract. Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded by a constant k. The min- imum cost of such a dominating broadcast is the dominating k-broadcast number. We present a unified upper bound on this parameter for any value of k, in terms of both k and the order of the graph. We also prove that the decision problem of determine if a graph has a dominating k-broadcast function with cost at most s is NP-complete. Keywords. Broadcast, domination, graphs. Domination in graphs has shown as a extremely fruitful concept, since it was originally defined in the late fifties [1] and named in the early sixties [10]. A dominating set of a graph G is a vertex set S such that any vertex not in S has at least one neighbor in S . Multiple variants of domination have been defined over the past fifty years, putting the focus on different aspects. One of them is the idea of broadcasting, firstly introduced in [9] and taken up more recently in [5]. This model reflects the idea of several broadcast stations, with associated transmission powers, that can broadcast messages to places at distance greater than one. We recall the formal definition from [5]. Definition 1. A broadcast on G is a function f : V (G) →{0, 1, . . . , diam(G)}. A vertex v ∈ V (G) with f (v) > 0 is a f -dominating vertex and the vertex set V + f = {v ∈ V (G): f (v) > 0} is the f -dominating set. An f -dominating vertex v is said to f -dominate every vertex u with d(u, v) ≤ f (v). A dominating broadcast on G is a broadcast f such that every vertex in G is f -dominated. The cost of a f -dominating broadcast is σ(f )= ∑ v∈V + f f (v) and the dominating broadcast number γ B (G) is the minimum value of σ(f ) over all dominating broadcast of G.  Partially supported by MINECO MTM2014-60127-P, MTM2015-63791-R (MINECO/FEDER), Gen.Cat. DGR2014SGR46 and Junta de Andaluc´ ıa FQM305