The minmin coalition number in graphs

TL;DR

This paper introduces the minmin coalition number in graphs, characterizes its bounds, and provides algorithms and conditions for specific cases, advancing understanding of coalition partitions in graph theory.

Contribution

It defines the minmin coalition number, characterizes graphs with extremal values, and offers polynomial-time algorithms and conditions for particular graph classes.

Findings

Bounds for c_min(G): 2 ≤ c_min(G) ≤ n

Characterization of graphs with c_min(G) = n

Polynomial-time algorithm for c_min(G) = 2

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 $X_j \in \mathcal{X}$. %The coalition number ${C}(G)$ is the maximum cardinality of a $c$-partition of $G$. Let ${\cal A} = \{A_1,\ldots,A_r\}$ and ${\cal B}= \{B_1,\ldots, B_s\}$ be two partitions of $V(G)$. Partition ${\cal B}$ is a refinement of partition ${\cal A}$ if every set $B_i \in {\cal B} $ is either equal to, or a proper subset of, some set $A_j \in {\cal A}$. Further if ${\cal A} \ne {\cal B}$, then ${\cal B}$ is a proper refinement of ${\cal A}$. Partition ${\cal A}$ 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.