Interpolation theorems for the (r, s)-domination number of spanning trees

If G is a graph without isolated vertices, and if rand s are positive integers, then the (r, s)-domination number 'Yr,s(G) of G is the cardinality of a smallest vertex set D such that every vertex not in D is within distance r from some vertex in D, while every vertex in D is within distance s from another vertex in D. This generalizes the total domination number 'Yt(G) = 'Yl,l(G). Let T( G) denote the set of all spanning trees of a connected graph G. We prove that 'Yr,s(T(G)) is a set of consecutive integers for every connected graph G of order at least two when s 2': 2r + 1. This is not true if 1 :::; s :::; 2r -1, and for s = 2r the problem is open. We prove that 'Yr,2r(T( G)) is a set of consecutive integers for r = 1 and we conjecture this also holds for r 2': 2. We also prove that 'Yr,s(T(G)) is a set of consecutive integers for every 2-connected graph G and for any two positive integers rand s. Let G be a simple undirected graph with vertices V(G) and edges E(G). The neighbourhood of a vertex v in G is NG(v) = {u E V(G) : uv E E(G)} and the closed neighbourhood is N G [v] = N G ( v) U { v }. For a connected graph G, let dG ( v, u) denote the distance between vertices v and u in G. If S is a set of vertices of G and v is a vertex of G, then dG ( v, S) denotes the distance between v and S, the shortest distance between v and a vertex of S. Let rand s be two positive integers. A vertex set D of a graph G is an (r, )-set of G if dG(v, D) :::; r for every v E V(G) D. Similarly, a subset D of V(G) is Australasian Journal of Combinatorics 17(1998), pp.99-107 a (-,s)-set of G if dc(u,D {u}) ::; s for every u E D. A subset D of V(G) is an (r, s) -dominating set of G if D is both an (r, )-set and a (-, s )-set of G. The cardinality of a minimum (r, s)-dominating set in G is called the (r, s)-domination number of G and is denoted bY'Yr,s(G). Note that this parameter is only defined for graphs without isolated vertices and if G is a graph without isolated vertices, then 'Yr,s (G) ~ 2. The (r, s )-domination number introduced by Mo and Williams [11] is related to other graphical parameters. In particular, the (1, I)-domination number "11,1 (G) of a graph G is the total domination number "It ( G) of G defined by Cockayne, Dawes and Hedetniemi [1]. The (r, r)-domination number was studied in [8] as the total P s ~ 2r + 1 and any shortest path joining x to a vertex of D {x} contains a vertex y for which dc(y, D) > r, which contradicts the fact that D is a distance r-dominating set in G. In addition, since D is a minimum distance r-dominating set of G, D is a minimum (r, s)-dominating set of G and therefore "ir,s(G) = 'Yr(G) = max{2, 'Yr(G)}. 0 Theorem 1. The (r, s)-domination number "ir,s is an interpolating function if s ~ 2r + 1.