Total mixed domination in graphs

TL;DR

This paper introduces the concept of total mixed domination in graphs, establishing bounds, relations with other graph parameters, and calculating this number for specific graph classes.

Contribution

It initiates the study of total mixed domination number, providing bounds, relations to diameter, and exact values for special graphs, expanding domination theory.

Findings

Established bounds in terms of order and total domination numbers

Linked total mixed domination number to graph diameter

Calculated total mixed domination numbers for specific graph classes

Abstract

For a graph $G=(V,E)$, we call a subset $ S\subseteq V \cup E$ a total mixed dominating set of $G$ if each element of $V \cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $\gamma_{tm}(G)$ of $G$ is the minimum cardinality of a total mixed dominating set of $G$. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number of a graph and its diameter. Studing of this number in trees is our next work. Also we show that the total mixed domination number of a graph is equale to the total domination number of a graph which is obtained by the graph. Giving the total mixed domination numbers of some special graphs is our last work.