On the local metric dimension of a lollipop graph, a web graph, and a friendship graph
A N Cahyabudi, Tri Atmojo Kusmayadı · Journal of Physics Conference Series · 2017
Let G be a simple connected graph with the vertex set V ( G ) and the edge set E ( G ). The distance between two vertices u and v , denoted by d ( u , v ), is the length of a shortest u − v path. If W = { w 1 , w 2 , w 3 , …, w n } is a finite set of vertices of G and v ∈ V ( G ), then the representation of v with respect to W is an ordered n -tuple r ( v | W ) = ( d ( v , w 1 ), d ( v , w 2 ), d ( v , w 3 ), …, d ( v , w n )). The set W is called a local metric generator for G if every two adjacent vertices of G has distinct representations. A minimum local metric generator is called a local metric basis for G and its cardinality is called the local metric dimension of G . A lollipop graph L m , n for m ≥ 3 and n ≥ 2 is the graph obtained by joining a complete graph K m to a path graph P n with a bridge. A web graph W n for n ≥ 3 is a generalized prism graph Y n +1,3 with the edge of the outer cycle removed. A friendship graph f n for n ≥ 2 is a graph constructed by joining n copies of the cycle graph C 3 with a common vertex. In this paper, we determine the local metric dimension of a lollipop graph, a web graph, and a friendship graph.