Symmetry properties of resolving sets and metric bases in hypercubes

A bridge between the minimal doubly resolving set problem in (folded) hypercubes and the coin weighing problem

“…Then Hamming graph H n,q is a vertex-transitive graph. By the Lemma 2.4(b), we have the following corollary, which was proved for q = 2 on the metric dimension problem in [26].…”

“…In addition, our upper bound of Ψ(Q 28 ) is better than the upper bound of β(Q 28 ) that founded by IPBS (see Table 3). What is more, when 29 ≤ n ≤ 90, our upper bounds of Ψ(Q n ) are not more than the upper bounds of β(Q n ) that is calculated by a dynamic programming (DP) procedure in [26] (see Recall that Chartrand et al [7] and Kratica et al [20] have given the 0-1 integer linear programming formulations for the metric dimension problem and the minimal doubly resolving set problem respectively. Using the similar method, we give the 0-1 integer linear programming formulations for computing φ(G, s).…”

“…Besides the genetic algorithm and the variable neighborhood search algorithm as mentioned previously, some specially algorithms are designed for the metric dimension and minimal doubly resolving set problem in hypercubes. For example, Nikolić et al [26] designed a greedy algorithm and a dynamic programming procedure for the metric dimension of hypercube by using the symmetry property of resolving sets to reduce the size of the feasible solution set. Hertz [15] designed an IP-based swapping algorithm for the metric dimension and minimal doubly resolving set problem in hypercubes.…”

On Metric Dimensions of Hypercubes

“…To this end, we will need the following two results. We must remark that the first of next two lemmas already appeared in [17]. We include its proof here by completeness of our paper.…”

“…The metric dimension of hypercube graphs has attracted the attention of several researchers from long ago. For instance, the work of Lindström [16] is probably one of the oldest ones, and for some recent ones we suggest the works [7,17,22]. Surprisingly, for other related invariants, there has been comparatively little research on hypercube graphs, although one can find some interesting recent results on this topic, such as those that appear in [5,7].…”