Singleton Coalition Graph Chains
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Let G be graph with vertex set V and order n=|V|. A coalition in G is a combination of two distinct sets, A⊆ V and B⊆ V, which are disjoint and are not dominating sets of G, but A∪ B is a dominating set of G. A coalition partition of G is a partition 𝒫={S_1,…,S_k} of its vertex set V, where each set S_i∈𝒫 is either a dominating set of G with only one vertex, or it is not a dominating set but forms a coalition with some other set S_j ∈𝒫. The coalition number C(G) is the maximum cardinality of a coalition partition of G. To represent a coalition partition 𝒫 of G, a coalition graph (G, 𝒫) is created, where each vertex of the graph corresponds to a member of 𝒫 and two vertices are adjacent if and only if their corresponding sets form a coalition in G. A coalition partition 𝒫 of G is a singleton coalition partition if every set in 𝒫 consists of a single vertex. If a graph G has a singleton coalition partition, then G is referred to as a singleton-partition graph. A graph H is called a singleton coalition graph of a graph G if there exists a singleton coalition partition 𝒫 of G such that the coalition graph (G,𝒫) is isomorphic to H. A singleton coalition graph chain with an initial graph G_1 is defined as the sequence G_1→ G_2→ G_3→⋯ where all graphs G_i are singleton-partition graphs, and (G_i,Γ_1)=G_i+1, where Γ_1 represents a singleton coalition partition of G_i. In this paper, we address two open problems posed by Haynes et al. We characterize all graphs G of order n and minimum degree δ(G)=2 such that C(G)=n and investigate the singleton coalition graph chain starting with graphs G where δ(G)≤ 2.
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