Minimal graphs with disjoint dominating and total dominating sets

TL;DR

This paper investigates the properties of minimal graphs that have disjoint dominating and total dominating sets, providing a characterization of such graphs without loops.

Contribution

It offers a new characterization of minimal DTDP-graphs without loops, advancing understanding of their structural properties.

Findings

Characterization of minimal DTDP-graphs without loops

Identification of properties distinguishing these graphs

Extension of existing theories on dominating sets

Abstract

A graph $G$ is a DTDP-graph if it has a pair $(D,T)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $T$ is a total dominating set of $G$. Such graphs were studied in a number of research papers. In this paper we study further properties of DTDP-graphs and, in particular, we characterize minimal DTDP-gaphs without loops.