Total Coalitions in Claw-Free Cubic Graphs Containing Double-Bonded Triangle-Units

A set S of vertices in an isolate-free graph G is a total dominating set if every vertex of G is adjacent to some other vertex in S. A total coalition in G consists of two disjoint sets of vertices X and Y of G, neither of which is a total dominating set but whose union X∪Y is a total dominating set of G. Such sets X and Y are said to form a total coalition. A total coalition partition in G is a vertex partition Ψ={V1,V2,…,Vk} such that for all i∈[k], the set Vi forms a total coalition with another set Vj for some j, where j∈[k]\{i}. We emphasize that none of the sets in Ψ is a total dominating set of G. The total coalition number Ct(G) in G equals the maximum order of a total coalition partition in G. We study total coalitions in claw-free cubic graphs with certain structural properties, namely, graphs containing double-bonded triangle-units, that is, two vertex disjoint triangles joined by two edges.