Computing the metric dimension of truncated wheels

• Main Authors: Garces, Ian June L, Rosario, Jose B
• Format: text
• Published: Archīum Ateneo 2015
• Subjects: resolving set; metric dimension; truncated wheel; Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/50; https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1049&context=mathematics-faculty-pubs
• Institution: Ateneo De Manila University

For an ordered subset W = {w1, w2, w3, . . . , wk} of vertices in a connected 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, w3), . . . , d(v, wk)). The set W is called a resolving set of G if r(u|W) = r(v|W) implies u = v. The metric dimension of G, denoted by β(G), is the minimum cardinality of a resolving set of G. Let n ≥ 3 be an integer. A truncated wheel, denoted by TWn, is the graph with vertex set V (TWn) = {a} ∪ B ∪ C, where B = {bi : 1 ≤ i ≤ n} and C = {cj,k : 1 ≤ j ≤ n, 1 ≤ k ≤ 2}, and edge set E(TWn) = {abi : 1 ≤ i ≤ n} ∪ {bici,k : 1 ≤ i ≤ n, 1 ≤ k ≤ 2} ∪ {cj,1cj,2 : 1 ≤ j ≤ n} ∪ {cj,2cj+1,1 : 1 ≤ j ≤ n}, where cn+1,1 = c1,1. In this paper, we compute the metric dimension of truncated wheels.