# Quasi-total Roman domination in graphs

## TL;DR

This paper introduces the quasi-total Roman domination number in graphs, exploring its properties and computational aspects, expanding the understanding of domination functions in graph theory.

## Contribution

It defines the quasi-total Roman domination number and initiates the study of its combinatorial and computational properties.

## Key findings

- Defined the quasi-total Roman domination number.
- Analyzed its combinatorial properties.
- Explored computational aspects.

## Abstract

A quasi-total Roman dominating function on a graph $G=(V, E)$ is a function $f : V \rightarrow \{0,1,2\}$ satisfying the following: - every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) =2$, and - if $x$ is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then $f(x)=1$. The weight of a quasi-total Roman dominating function is the value $\omega(f)=f(V)=\sum_{u\in V} f(u)$. The minimum weight of a quasi-total Roman dominating function on a graph $G$ is called the quasi-total Roman domination number of $G$. We introduce the quasi-total Roman domination number of graphs in this article, and begin the study of its combinatorial and computational properties.