Singleton coalition graph chains

Let G be a 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 their union 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 CG(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 CG(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 CG(G_i, _1)=G_i+1 , where _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 . Additionally, we investigate the singleton coalition graph chain starting with graphs G , where δ (G)≤ 2 .