Abstract

Let G be a connected nontrivial graph with vertex set V(G) and edge set E(G). The distance between two vertices u and v in G is the shortest path length between u and v denoted d(u, v). Let W = {w1, w2, …, wk} be a subset of V(G). The representation of a vertex u with respect to W is a sequential pair of distances between u and all vertices in W, where u is a vertex in G. The set of W is called the resolving set if the representation of each vertex is different to W. Resolving set with a minimum cardinality called the metric basis and the number of element from some basis is called the metric dimension, denoted by dim(G). In this paper, we determine the metric dimension of amalgamation of sunfiower and lollipop graph (SFn, vi) ✱ (Lm, p, up) and caveman graph C(n, m). The results show that the metric dimension of amalgamation of sunflower and lollipop graph is dim((SFn, vi) ✱ (Lm, p, up)) = m + 1 for n = 3, 4, …, 7; for n ≥ 8, and the metric dimension of caveman graph is dim(C(n, m)) = n for m = 3, 4; dim(C(n, m)) = (m – 4)n for m ≥ 5.

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