The minmin coalition number in graphs
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A set S of vertices in a graph G is a dominating set if every vertex of V(G) ∖ S is adjacent to a vertex in S. A coalition in G consists of two disjoint sets of vertices X and Y of G, neither of which is a dominating set but whose union X ∪ Y is a dominating set of G. Such sets X and Y form a coalition in G. A coalition partition, abbreviated c-partition, in G is a partition 𝒳 = {X_1,…,X_k} of the vertex set V(G) of G such that for all i ∈ [k], each set X_i ∈𝒳 satisfies one of the following two conditions: (1) X_i is a dominating set of G with a single vertex, or (2) X_i forms a coalition with some other set X_j ∈𝒳. maximum cardinality of a c-partition of G. Let A = {A_1,…,A_r} and B= {B_1,…, B_s} be two partitions of V(G). Partition B is a refinement of partition A if every set B_i ∈ B is either equal to, or a proper subset of, some set A_j ∈ A. Further if A B, then B is a proper refinement of A. Partition A is a minimal c-partition if it is not a proper refinement of another c-partition. Haynes et al. [AKCE Int. J. Graphs Combin. 17 (2020), no. 2, 653–659] defined the minmin coalition number c_min(G) of G to equal the minimum order of a minimal c-partition of G. We show that 2 ≤ c_min(G) ≤ n, and we characterize graphs G of order n satisfying c_min(G) = n. A polynomial-time algorithm is given to determine if c_min(G)=2 for a given graph G. A necessary and sufficient condition for a graph G to satisfy c_min(G) ≥ 3 is given, and a characterization of graphs G with minimum degree 2 and c_min(G)= 4 is provided.
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