Fundamental Dominations in Graphs

TL;DR

This paper introduces a unified framework for nine fundamental domination concepts in graphs, analyzes their relationships, and interprets them via total graphs, highlighting the foundational nature of the original domination concept.

Contribution

It unifies nine domination variations under a common approach, establishes bounds among their numbers, and links them to total graphs, emphasizing the primacy of the original domination concept.

Findings

At most five of the nine domination numbers differ in a connected graph.

Inequalities between these five numbers are established.

Fundamental domination concepts are interpreted through total graphs.

Abstract

Nine variations of the concept of domination in a simple graph are identified as fundamental domination concepts, and a unified approach is introduced for studying them. For each variation, the minimum cardinality of a subset of dominating elements is the corresponding fundamental domination number. It is observed that, for each nontrivial connected graph, at most five of these nine numbers can be different, and inequalities between these five numbers are given. Finally, these fundamental dominations are interpreted in terms of the total graph of the given graph, a concept introduced by the second author in 1965. It is argued that the very first domination concept, defined by O. Ore in 1962 and under a different name by C. Berge in 1958, deserves to be called the most fundamental of graph dominations.