# Small domination-type invariants in random graphs

## TL;DR

This paper investigates the behavior of a generalized domination invariant, called the $c$-self domination number, in random graphs, extending classical domination concepts and analyzing their properties for small values of $c$.

## Contribution

It introduces the $c$-self domination number as a unifying generalization of domination, total domination, and Roman domination, and studies its behavior in random graphs for small $c$.

## Key findings

- Analyzes the $c$-self domination number in Erdős–Rényi random graphs.
- Provides bounds and asymptotic behavior for small $c$ values.
- Connects the generalized invariant to classical domination parameters.

## Abstract

For $c\in \mathbb{R}^{+}\cup \{\infty \}$ and a graph $G$, a function $f:V(G)\rightarrow \{0,1,c\}$ is called a $c$-self dominating function of $G$ if for every vertex $u\in V(G)$, $f(u)\geq c$ or $\max\{f(v):v\in N_{G}(u)\}\geq 1$ where $N_{G}(u)$ is the neighborhood of $u$ in $G$. The minimum weight $w(f)=\sum _{u\in V(G)}f(u)$ of a $c$-self dominating function $f$ of $G$ is called the $c$-self domination number of $G$. The $c$-self domination concept is a common generalization of three domination-type invariants; (original) domination, total domination and Roman domination. In this paper, we study a behavior of the $c$-self domination number in random graphs for small $c$.