A minimum metric basis is a minimum set W of vertices of a graph G(V , E) such that for every pair of vertices u and v of G, there exists a vertex w ∈ W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. The honeycomb and hexagonal networks are popular mesh-derived parallel architectures. Using the duality of these networks we determine minimum metric bases for hexagonal and honeycomb networks.

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The most popular bounded-degree derivative network of the hypercube is the butterfly network. The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly-like architectures. We identify a new topological representation of butterfly and Benes networks.The minimum metric dimension problem is to find a minimum set of vertices of a graph G(V , E) such that for every pair of vertices u and v of G, there exists a vertex w with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. It is NP-hard in the general sense. We show that it remains NP-hard for bipartite graphs. The algorithmic complexity status of this NP-hard problem is not known for butterfly and Benes networks, which are subclasses of bipartite graphs. By using the proposed new representations, we solve the minimum metric dimension problem for butterfly and Benes networks. The minimum metric dimension problem is important in areas such as robot navigation in space applications.

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