Radius-forcing sets in graphs

Let G be a connected graph of order p and let 0 ::f. s ~ V ( G). Then S is a rad(G)-forcing set (or a radius-forcing set of G) if, for each v E V(G), there exists v' E S with dc(v, v') ?: rad(G). The cardinality of a smallest radius-forcing set of G is called the radius-forcing number of G and is denoted by rf(G). A graph G is called a randomly k-forcing graph for a positive integer k if every k-subset of V(G) is a radius-forcing set of G. We investigate the value of rf(G) for various graphs G, and obtain some general bounds, and we characterize graphs for which rf achieves the values of 1, 2, p-1, and p, respectively. We establish the NPcompleteness of the calculation of rf for arbitrary graphs, and conclude with an investigation of k-randomly forcing graphs. 1. Introductory definitions and examples Let G be a connected graph of order p and vertex set V (G). Suppose that the vertices of G represent p facilities in which essential data or materials are storeable (for example, warehouses, rooms, computers in an information network). Two vertices in G are joined by an edge if the corresponding facilities are linked or adjacent or are somehow "close" to each other. Suppose that it has been determined that, for some kEN, if a disaster or failure of some kind occurs at a facility (represented by a vertex v, say), then all facilities represented by vertices at distance at most k 1 from v will be jeopardized. The problem at hand now is to select the smallest possible subset of V (G) so that, if our essential material is stored in the facilities corresponding to this subset, then our system, in the most economical way, has the property that our material, or information, is retrievable from somewhere in the system even in the case when an arbitrary facility fails. One option, of course, is to design G to have radius at least k and to store all essential data in each facility, but this is an expensive option. However, if rad( G) ?: k and if S is a smallest subset of V(G) with the property that, for each w E V(G), there exists w' E S such that dc ( w, w') ?: k, then selecting the lSI facilities represented by S as the set of facilities at which to store our essential data will produce a choice that may be considerably cheaper, but which still provides the required security. In this paper, we will consider the case where rad( G) = k. Australasian Journal of Combinatorics 17(1998), pp.39-49 Let G be a (connected) graph, a E V(G) and A, S ~ V(G). We define the generalized S-eccentricity of a in G, eG(a, S), by eG(a, S) = max{dG(a, s); s E S}. If w is a vertex in S for which eG(a, S) = dG(a, w), we will call w an 8-eccentric vertex of a. We define rad(A, S, G), the radius of A with respect to 8 in G, by rad(A,S,G) = min{eG(a,S); a E A}. If S is such that rad(A, S, G) = rad( G), then S is called an A-rad(G}-forcing set; the size of a smallest A-rad( G)-forcing set is denoted by rf(A, G) and called the A-rad(G}-forcing number. If rad(V(G), S, G) = rad(G), then (briefly) 8 is a rad(G)forcing set, or simply a radius-forcing set if no ambiguity is possible; the size of a smallest rad( G)-forcing set, denoted by rf( G), is called the radius-forcing number of G. Also, we abbreviate rad(V(G), S, G) by rad(S, G). (Notice that rf(G) can be seen as the smallest number of vertices in a subset 8 of V(G) such that each vertex of G is at distance at least rad(G) from some vertex in S.) In [4], Fajtlowicz introduced the class of graphs called r-ciliates and the following notion of r-criticality. Definition 1. For a, bEN with b ~ 3, let Cb,a be a graph obtained from b disjoint copies of Pa+1 by linking together one end-vertex of each in a cycle Cb• For r, a E N with r 2: a, the graphs C2a,r-a are called r-ciliates. A graph is r-critical if it has radius r and every proper induced connected subgraph has radius strictly smaller than r. Finally, for a connected graph G of radius r, we define the graph G* to be the graph given by V(G*) = V(G) and uv E E(G*) if and only if dG(u, v) 2: r. Notice that this graph G* provides a link between total domination and radiusforcing number since, by the definition of rf, '"Yt(G*) = rf(G). Furthermore, it is not difficult to see that G* = Grad(G)-l. (This graph is a generalization, in a sense, of the antipodal graph A(G) of a graph G defined by R. R. Singleton [6], where A(G) C G* and uv E E(A(G)) if and only if dG(u, v) = diam{G).)