On the multipacking number of grid graphs

In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph G is denoted γb(G). The dual of this problem is called multipacking : a multipacking is a set M ⊆ V (G) such that for any vertex v and any positive integer r, the ball of radius r around v contains at most r vertices of M . The maximum size of a multipacking in a graph G is denoted mp(G). Naturally mp(G) ≤ γb(G). Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal. Introduction Given a graph G with vertex set V and edge set E, a dominating broadcast of G is a function f from V to N such that for any vertex u in V , there is a vertex v in V with f(v) positive and greater than the distance from u to v. Define the ball of radius r around v by Nr(v) = {u : d(u, v) ≤ r}. Thus a dominating broadcast is a cover of the graph with balls of several positive radii. The cost of a dominating broadcast f is ∑ v∈V f(v) and the minimum cost of a dominating broadcast in G, its broadcast number, is denoted γb(G). Remark. One may consider the cost to be any function of the powers (for example the sum of the squares), see e.g. [10]. We shall stick to the classical convention of linear cost. The dual problem of broadcast domination is multipacking. A multipacking in a graph G is a subset M of its vertices such that for any positive integer r and any vertex v in V , the ball of radius r centred at v contains at most r vertices of M . The maximum size of a multipacking of G, its multipacking number, is denoted mp(G). We may write γb and mp when the graph in question is clear from context or unimportant. Broadcast domination was introduced by Erwin [7, 8] in his doctoral thesis in 2001. Multipacking was then defined in Teshima’s Master’s Thesis [13] in