Introduction to total coalitions in graphs

Let \(G\) be a graph with vertex set \(V\). Two disjoint sets \(V_1, V_2\subseteq V\) are called a total coalition in \(G\), if neither \(V_1\) and \(V_2\) is a total dominating set of \(G\) but \(V_1\cup V_2\) is a total dominating set. A total coalition partition of \(G\) is a vertex partition \(\...

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Bibliographic Details
Published in:arXiv.org 2022-12
Main Authors: Alikhani, Saeid, Bakhshesh, Davood, Golmohammadi, Hamidreza
Format: Article
Language:English
Subjects:
Apexes
Graph theory
Partitions (mathematics)
Online Access:Get full text
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Summary:Let \(G\) be a graph with vertex set \(V\). Two disjoint sets \(V_1, V_2\subseteq V\) are called a total coalition in \(G\), if neither \(V_1\) and \(V_2\) is a total dominating set of \(G\) but \(V_1\cup V_2\) is a total dominating set. A total coalition partition of \(G\) is a vertex partition \(\pi=\{V_1,V_2,\ldots, V_k\}\) such that no set of \(\pi\) is a total dominating set but each set \(V_i\in \pi\) forms a total coalition with another set \(V_j\in \pi\). The maximum cardinality of a total coalition partition of \(G\) is called the total coalition number of \(G\), denoted by \(TC(G)\). In this paper, we initiate the study of the total coalition in graphs and its properties.
ISSN:2331-8422