On graphs of order m with dominating partition dimension m − 3

Let Γ(V,E) be a simple connected graph and Q = {W1, ⋯, Wq} be a partition of V(Γ). The partition representation of a vertex j with respect to Q, denoted by pr(j|Q), is the vector (d(j,W1), d(j,W2), ⋯, d(j,Wq), where d(j,Wi) denotes the distance between vertex j and set Wi. The partition Q is called a resolving partition of Γ if any two vertices j,k ∈ V(Γ) have distinct representations with respect to Q. The partition dimension of Γ, denoted by ζp(Γ), is the least cardinality of a resolving partition of graph Γ. The partition Q is dominating if for each vertex j of Γ, there is an Wi such that d(j,Wi) = 1. If the partition Q are resolving as well as dominating then we call Q a resolving dominating partition of Γ. The minimum cardinality of a resolving dominating partition of Γ is called the dominating partition dimension of Γ and it is denoted by δp(Γ). For given positive integers m,b with b ≤ m, characterizing all graphs of order m with dominating partition dimension b is a tough issue. There are still few results concerning this problem, in particular for b equal to 2, 3, m − 2, m − 1, or m. In this paper, we shall focus on studying the graphs Γ of order m with dominating partition dimension m − 3. We will give some graphs of order m and diameter 2 with dominating partition dimension m − 3.