Nordhaus–Gaddum bounds for locating domination

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 locatingdominating 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 G. 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.

The difference between the metric dimension and the determining number of a graph

“…By Theorem 4.3, kðGÞ 6 n À CðGÞ for every twin-free graph G. Further, Theorem 7 of [25] gives jkðGÞ À kðGÞj 6 1 and so kðGÞ 6 kðGÞ þ 1 6 n À CðGÞ þ 1 since G is also twin-free. Therefore, kðGÞ 6 minfn À CðGÞ; n À CðGÞ þ 1g.…”

On global location-domination in graphs

“…Proof. The values of the location-domination number of all these families, except the wheels, are already known (see [1,12,18]). Next, let us calculate the values of the location-domination number for the wheels and for the complements of all these families and also, from the results previously proved, the global location-domination number of them.…”

“…Notice also that a locating-dominating set is both a locating set and a dominating set, and thus β(G) ≤ λ(G) and γ(G) ≤ λ(G). LDcodes and the location-domination parameter have been intensively studied during the last decade; see [1,2,5,8,12] A complete and regularly updated list of papers on locating dominating codes is to be found in [14].…”

Locating domination in bipartite graphs and their complements

“…Certainly, every LD-set of a non-connected graph G is the union of LD-sets of its connected components and the location-domination number is the sum of the location-domination number of its connected components. Both, LD-codes and the location-domination parameter have been intensively studied during the last decade; see [1,2,3,5,6,8,9,10]. A complete and regularly updated list of papers on locating-dominating codes is to be found in [11].…”