The complement metric dimension of particular tree

Let G be a connected graph with vertex set V(G) and edge set E(G). The distance between vertices u and v in G is denoted by d(u,v), which serves as the shortest path length from u to v. Let W = {w_hw₂,...,w_k} ⊆ V(G) be an ordered set, and v is a vertex in G. The representation of v with respect to W is an ordered set k-tuple, r(v|W) = (d(v,w₁),d(v,w₂),...,d(w_k)). The set Wis called a complement resolving set for G if there are two vertices u,v⊆V(G)\W, such that r(u|W)=r(v|W). A complement basis of G is the complement resolving set containing maximum cardinality. The number of vertices in a complement basis of G is called complement metric dimension of G, which is denoted by $\begin{equation}\overline{d i m}\end{equation}$ (G). In this paper, we examined complement metric dimension of particular tree graphs such as caterpillar graph (C_mn), firecrackers graph (F_mn), and banana tree graph (B_m,_n). We got $\begin{equation}\overline{d i m}\end{equation}$ = m(n+1)-2 for m>1 and n>2, $\begin{equation}\overline{d i m}\end{equation}$ = m(n+2)-2 for m>1 and n>2, and $\begin{equation}\overline{d i m}\end{equation}$ = m(n+1)-1 if m>1 and n>2.