Possible Cardinalities for Locating-Dominating Codes in Graphs
Australasian Journal of Combinatorics, Vol. 34, pp. 23-32, 2006.
Irène CHARON, Olivier HUDRY & Antoine LOBSTEIN
Centre National de la Recherche Scientifique
Ecole Nationale Supérieure des Télécommunications
46 rue Barrault, 75634 Paris cédex 13, France
{charon, hudry, lobstein}@infres.enst.fr
Abstract. Consider a connected undirected graph G=(V,E), a subset C of vertices, and an integer r greater than or equal to 1; for any vertex v in V, let Br(v) denote the ball of radius r centered at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v in V $\setminus$ C, the sets Br(v) $\cap$ C are all nonempty and different, then we call C an r-locating-dominating code.
It is known that the cardinality of a minimum r-locating-dominating code C in any connected undirected graph G having a given number, n, of vertices satisfies the inequalities |C| $\leq$ n-1 and |C|+2|C| $\geq$ n+1, and that these lower and upper bounds can be achieved. Here, we prove that any in-between value can also be reached by |C|.