Non-isolated Resolving Sets of certain Graphs Cartesian Product with a Path

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Let G be a connected, simple, and finite graph. For an ordered subset W = {w1, w2, · · ·, wk} of vertices in a graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v|W ) = (d(v, w1), d(v, w2), · · ·, d(v, wk)). The set W is called a resolving set for G if every vertex of G has a distinct representation. The minimum cardinality of W is called the metric dimension of G, denoted by dim(G). If the induced subgraph has no isolated vertices, then W is called a non-isolated resolving set. The minimum cardinality of non-isolated resolving set of G is called the non-isolated resolving number of G, denoted by nr(G). In this paper, we consider that is a graph obtained from Cartesian product between a connected graph H and a path Pn. We determine , for some classes of H, including cycles, complete graphs, complete bipartite graphs, and friendship graphs.