On (local) metric dimension of graphs with m-pendant points

An ordered set of vertices S is called as a (local) resolving set of a connected graph G = (V_G, E_G) if for any two adjacent vertices s ≠ t ∈ V_G have distinct representation with respect to S, that is r(s | S) ≠ r(t | S). A representation of a vertex in G is a vector of distances to vertices in S. The minimum (local) resolving set for G is called as a (local) basis of G. A (local) metric dimension for G denoted by dim(G), is the cardinality of vertices in a basis for G, and its local variant by dim_l (G). Given two graphs, G with vertices s₁, s₂, ⋯, s_p and edges e₁, e₂, ⋯, e_q , and H. An edge-corona of G and H, G⋄H is defined as a graph obtained by taking a copy of G and q copies of H and for each edge e_j = s_is_h of G joining edges between the two end-vertices s_i, s_h of e_j and each vertex of j-copy of H. In this paper, we determine and compare between the metric dimension of graphs with m-pendant points, G⋄mK₁, and its local variant for any connected graph G. We give an upper bound of the dimensions.