Total coalitions in graphs.

AbstractWe define a total coalition in a graph <italic>G</italic> as a pair of disjoint subsets <italic>A</italic>1,<italic>A</italic>2 ⊆ <italic>A</italic> that satisfy the following conditions: (a) neither <italic>A</italic>1 nor <italic>A</italic>2 constitutes a total dominating set of <italic>G</italic>, and (b) <italic>A</italic>1 ∪ <italic>A</italic>2 constitutes a total dominating set of <italic>G</italic>. A total coalition partition of a graph <italic>G</italic> is a partition ϒ = {<italic>A</italic>1,<italic>A</italic>2, . . . ,<italic>Ak</italic>} of its vertex set such that no subset of ϒ acts as a total dominating set of <italic>G</italic>, but for every set <italic>Ai</italic> ∈ ϒ, there exists a set <italic>Aj</italic> ∈ ϒ such that <italic>Ai</italic> and <italic>Aj</italic> combine to form a total coalition. We define the total coalition number of <italic>G</italic> as the maximum cardinality of a total coalition partition of <italic>G</italic>, and we denote it by <italic>Ct</italic>(<italic>G</italic>). The purpose of this paper is to begin an investigation into the characteristics of total coalition in graphs. [ABSTRACT FROM AUTHOR]