Nordhaus-Gaddum bounds for locating domination

TL;DR

This paper establishes tight Nordhaus-Gaddum bounds for the location, metric-location, and location-domination numbers in graphs, characterizing the extremal graph families for each case.

Contribution

It provides new bounds for these parameters and characterizes the graphs that attain these bounds, advancing understanding of graph locating and dominating sets.

Findings

Derived tight Nordhaus-Gaddum bounds for b, e, and l parameters.

Characterized extremal graph families for each bound.

Enhanced understanding of locating and dominating sets in graphs.

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 b-codes, e-codes and l-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 b, the metric-location-number e and the location-domination number l. Moreover, in each case, the graph family attaining the corresponding bound is characterized.