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

TL;DR

This paper explores the minimal doubly resolving set problem in hypercubes and folded hypercubes, establishing its equivalence to the coin weighing problem, disproving a conjecture, and providing new bounds and algorithms.

Contribution

It links the minimal doubly resolving set problem to the coin weighing problem, answers an open question, and introduces an efficient algorithm with new bounds.

Findings

Equivalence between the doubly resolving set problem in hypercubes and coin weighing problem.

Disproof of a conjecture on metric dimension in folded hypercubes.

NP-hardness of the problem in various graph classes.

Abstract

In this paper, we consider the minimal doubly resolving set problem in Hamming graphs, hypercubes and folded hypercubes. We prove that the minimal doubly resolving set problem in hypercubes is equivalent to the coin weighing problem. Then we answer an open question on the minimal doubly resolving set problem in hypercubes. We disprove a conjecture on the metric dimension problem in folded hypercubes and give some asymptotic results for the metric dimension and the minimal doubly resolving set problems in Hamming graphs and folded hypercubes by establishing connections between these problems. Using the Lindstr\"{o}m's method for the coin weighing problem, we give an efficient algorithm for the minimal doubly resolving set problem in hypercubes and report some new upper bounds. We also prove that the minimal doubly resolving set problem is NP-hard even restrict on split graphs, bipartite graphs and co-bipartite graphs.