Abstract
In this paper the strong metric dimension of generalized Petersen graphs GP(n, 2) is considered. The exact value is determined for the cases n = 4k and n = 4k + 2, while for n = 4k + 1 an upper bound of the strong metric dimension is presented.
Highlights
The strong metric dimension problem was introduced by Sebo and Tannier [13]
The exact value is determined for the cases n = 4k and n = 4k + 2, while for n = 4k + 1 an upper bound of the strong metric dimension is presented
For k = 1, using CPLEX solver on integer linear programming (ILP) formulation (1), we have proved that the set S from Lemma 1 is a strong metric basis of GP (6, 2), i.e. sdim(GP (6, 2)) = 6
Summary
The strong metric dimension problem was introduced by Sebo and Tannier [13]. This problem is defined in the following way. ([4]) If S ⊂ V is a strong resolving set of graph G, for every two maximally distant vertices u, v ∈ V , it must be u ∈ S or v ∈ S. We consider the strong metric dimension of the generalized Petersen graph GP (n, 2). The last two columns named ”condition” and ”shortest path” contain the condition under which the vertex in column three strongly resolves the pair in column two, and the corresponding shortest path, respectively. Vertices which strongly resolve pair (ui, vj) listed in Table 1 belong to the set S.
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