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 d i m(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 d i m = m(n+1)-2 for m>1 and n>2, d i m = m(n+2)-2 for m>1 and n>2, and d i m = m(n+1)-1 if m>1 and n>2.