The Metric Dimension of Graph with Pendant Edges

For an ordered set \( W = \{w_1, w_2, \dots, w_k\} \) of vertices and a vertex \( v \) in a connected graph \( G \), the representation of \( v \) with respect to \( W \) is the ordered \( k \)-tuple \( r(v|W) = (d(v, w_1), d(v, w_2), \dots, d(v, w_k)) \) where \( d(x,y) \) represents the distance between the vertices \( x \) and \( y \). The set \( W \) is called a resolving set for \( G \) if every two vertices of \( G \) have distinct representations. A resolving set containing a minimum number of vertices is called a basis for \( G \). The dimension of \( G \), denoted by \( \text{dim}(G) \), is the number of vertices in a basis of \( G \). In this paper, we determine the dimensions of some corona graphs \( G \odot K_1 \), \( G \odot \overline{K}_m \) for any graph \( G \) and \( m \geq 2 \), and a graph with pendant edges more general than corona graphs \( G \odot \overline{K}_m \).