On lower bounds for cardinalities of several separating-dominating codes in graphs

TL;DR

This paper investigates lower bounds for the size of separating-dominating codes in graphs, providing constructions of extremal graphs that achieve these bounds and characterizing all such extremal graphs.

Contribution

It introduces a general construction method for extremal graphs that minimize code size, reestablishes known bounds, and derives new bounds for various identification problems.

Findings

Provided constructions for extremal graphs achieving lower bounds

Reproved existing logarithmic lower bounds for certain codes

Characterized all graphs attaining these bounds

Abstract

In the literature, several different identification problems in graphs have been studied, the most widely studied such problems are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a "code" of the graph. The problems of determining such codes of minimum cardinality are all shown to be NP-hard. A typical line to attack such problems is, therefore, to provide bounds on the cardinalities of the studied codes. In this paper, we are interested in extremal graphs for lower bounds in terms of the order of the graph. For some codes, logarithmic lower bounds are known from the literature. We provide for eight different identification problems a general construction of extremal graphs minimizing the cardinality of a minimum code compared to the order of the graph. This enables us to reprove the existing logarithmic lower bounds, to establish such bounds for further codes, and to characterize all graphs attaining these bounds.