European journal of combinatorics

Vol. 36, p. 1-6

DOI: 10.1016/j.ejc.2013.04.009

Date of publication: 2014-02-01

Abstract:

A dominating set S of graph G is called metric-locating–dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S. If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S, then it is said to be locating–dominating. Locating, metric-locating–dominating and locating–dominating sets of minimum cardinality are called ß-codes, ¿-codes and ¿-codes, respectively. A Nordhaus–Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its complement View the MathML source. In this paper, we present some Nordhaus–Gaddum bounds for the location number ß, the metric-location–domination number ¿ and the location–domination number ¿. Moreover, in each case, the graph family attaining the corresponding bound is fully characterized.

A dominating set S of graph G is called metric-locating–dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S. If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S, then it is said to be locating–dominating. Locating, metric-locating–dominating and locating–dominating sets of minimum cardinality are called β-codes, η-codes and λ-codes, respectively. A Nordhaus–Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its complement View the MathML source. In this paper, we present some Nordhaus–Gaddum bounds for the location number β, the metric-location–domination number η and the location–domination number λ. Moreover, in each case, the graph family attaining the corresponding bound is fully characterized.]]>

European journal of combinatorics

Vol. 30, num. 7, p. 1585-1592

DOI: 10.1016/j.ejc.2009.03.008

Date of publication: 2009-10