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.

如果\(V(G) \setminus S\)的每个顶点都与S中的一个顶点相邻，则图G中的顶点集S是支配集。G中的联盟由G的两个不相交的顶点集X和Y组成，它们都不是支配集，但其并集\(X \cup Y\)是G的支配集。这样的集合X和Y在G中形成一个联盟。G中的联合划分，缩写为c -partition，是G的顶点集V ( G ) 的划分\({\mathcal {X}} = \{X_1,\ldots ,X_k\}\)，使得对于所有\(i \in [k]\)，每个集合\(X_i \in {\mathcal {X}}\)满足以下两个条件之一： (1) \(X_i\)是G的支配集，其中单个顶点，或 (2) \(X_i\)与其他集合\(X_j \in {\mathcal {X}}\)形成联盟。设\({{\mathcal {A}}} = \{A_1,\ldots ,A_r\}\)和\({{\mathcal {B}}}= \{B_1,\ldots , B_s\}\)是V ( G )的两个分区。分区\({{\mathcal {B}}}\)是分区\({{\mathcal {A}}}\)的细化，如果每个集合\(B_i \in {{\mathcal {B}}} \ )要么等于某个集合\(A_j \in {{\mathcal {A}}}\) ，要么是其真子集。此外，如果\({{\mathcal {A}}} \ne {{\mathcal {B}}}\)，则\({{\mathcal {B}}}\)是\({{ \mathcal {A}}}\)。如果分区\({{\mathcal {A}}}\)不是另一个c分区的适当细化，则它是最小c分区。海恩斯等人。 [AKCE 国际。 J.图组合。 17（2020），没有。 2, 653–659]定义了G的minmin 联盟数\(c_{\min }(G)\)等于G的最小c分区的最小阶。我们证明\(2 \le c_{\min }(G) \le n\)，并且我们描述满足\(c_{\min }(G) = n\)的n阶图 G 。给出一个多项式时间算法来确定给定图G是否\(c_{\min }(G)=2\)。给出了图G满足\(c_{\min }(G) \ge 3\)的充要条件，并给出了图G最小度为2和\(c_{\min }(G)的刻画= 4\)被提供。