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 h w 2 ,…,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,w 1 ),d(v,w 2 ),…,d(w k )). The set W is 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 dim ¯ ( 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 dim ¯ = m(n+1)-2 for m>1 and n>2, dim ¯ = m(n+2)-2 for m>1 and n>2, and dim ¯ = m(n+1)-1 if m>1 and n>2.