The complement metric dimension of particular tree

Abstract

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 = {whw2,…,wk} ⊆ 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(wk)). 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 (G). In this paper, we examined complement metric dimension of particular tree graphs such as caterpillar graph (Cmn), firecrackers graph (Fmn), and banana tree graph (Bm,n). We got = m(n+1)-2 for m>1 and n>2, = m(n+2)-2 for m>1 and n>2, and = m(n+1)-1 if m>1 and n>2.