The minmin coalition number in graphs

Abstract

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.