What question did this study set out to answer?

The research focuses on the conditions under which proper total dominating sets exist in graphs and their relationships with total domination numbers.

April 16, 2026Open Access

Realizability in Proper Total Domination in Graphs

Key Points

The research focuses on the conditions under which proper total dominating sets exist in graphs and their relationships with total domination numbers.
Defined key terminology related to total domination and proper total domination sets in graphs.
Analyzed relationships between total domination numbers and proper total domination numbers in graphs.
Demonstrated the construction of graphs for specified pairs of integers depicting domination characteristics.
Established that every graph with a proper total dominating set adheres to the inequality t(G) ≤ pt(G) ≤ pt(G).
Showed that for any integers a, b with 3 ≤ a ≤ b, connected graphs can be constructed with specific pt-domination parameters.
Identified relationships among integer triples a, b, c where 2 ≤ a ≤ b ≤ c, revealing their existence in connected graphs.

Abstract

A vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A set S of vertices of G is a total dominating set if every vertex of G is totally dominated by some vertex of S. The minimum cardinality of a total dominating set of G is its total domination number t(G) of G.A proper total dominating set (or pt-dominating set) of G is a total dominating set S where no two adjacent vertices of G are totally dominated by the same number of vertices of S. The minimum cardinality of a pt-dominating set of G is the pt-domination number pt(G), while the maximum cardinality of such a set is the upper pt-domination number pt(G).While every graph without isolated vertices has a total dominating set, this need not be the case for pt-dominating sets.For every graph G having a pt-dominating set, t(G) pt(G) pt(G).It is shown that for every pair a, b of integers with 3 a b, there exists a connected graph G with pt(G) = a and pt(G) = b.Furthermore, those triples a, b, c of integers with 2 a b c are studied for which there is a connected graph G with t(G) = a, pt(G) = b, and pt(G) = c.

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