Total restrained coalitions in graphs

TL;DR

This paper introduces the concept of total restrained coalition in graphs, exploring its properties and defining the total restrained coalition number as a measure of maximum partition size.

Contribution

It initiates the study of total restrained coalitions in graphs, defining key concepts and establishing foundational properties for this new area.

Findings

Defined total restrained coalition and related concepts.

Introduced the total restrained coalition number $C_{tr}(G)$.

Explored basic properties and potential bounds of $C_{tr}(G)$.

Abstract

A set $S\subseteq V$ in an isolate-free graph $G$ is a total restrained dominating set, abbreviated TRD-set, if every vertex in $V$ is adjacent to a vertex in $S$, and every vertex in $V\setminus S$ is adjacent to a vertex in $V\setminus S$. A total restrained coalition is made up of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a TRD-set but their union $X\cup Y$ is a TRD-set. A total restrained coalition partition of a graph $G$ is a partition $\Phi=\{V_1, V_2,\dots,V_k\}$ such that for all $i \in [k]$, the set $V_i$ forms a total restrained coalition with another set $V_j$ for some $j$, where $j\in [k]\setminus{i}$. The total restrained coalition number $C_{tr}(G)$ in $G$ equals the maximum order of a total restrained coalition partition in $G$. In this work, we initiate the study of total restrained coalition in graphs and its properties.