The minmin coalition number in graphs

A set S of vertices in a graph G is a dominating set if every vertex of \(V(G) \setminus S\) 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 \(X \cup Y\) 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 \({\mathcal {X}} = \{X_1,\ldots ,X_k\}\) of the vertex set V(G) of G such that for all \(i \in [k]\), each set \(X_i \in {\mathcal {X}}\) 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 \(X_j \in {\mathcal {X}}\). Let \({{\mathcal {A}}} = \{A_1,\ldots ,A_r\}\) and \({{\mathcal {B}}}= \{B_1,\ldots , B_s\}\) be two partitions of V(G). Partition \({{\mathcal {B}}}\) is a refinement of partition \({{\mathcal {A}}}\) if every set \(B_i \in {{\mathcal {B}}} \) is either equal to, or a proper subset of, some set \(A_j \in {{\mathcal {A}}}\). Further if \({{\mathcal {A}}} \ne {{\mathcal {B}}}\), then \({{\mathcal {B}}}\) is a proper refinement of \({{\mathcal {A}}}\). Partition \({{\mathcal {A}}}\) 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 \(2 \le c_{\min }(G) \le n\), 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 \(c_{\min }(G) \ge 3\) is given, and a characterization of graphs G with minimum degree 2 and \(c_{\min }(G)= 4\) is provided.