Metric Dimension of a Direct Product of Three Complete Graphs: The Middle Cone Family

TL;DR

This paper investigates the metric dimension and location-total-domination number of a specific family of direct products of three complete graphs from the middle cone, generalizing previous work on isomorphic factors.

Contribution

It determines the metric dimension and resolving sets for the middle cone family of three complete graphs with different orders, extending existing methods.

Findings

Explicit minimum resolving sets described.

A hypergraph-based criterion for resolving sets established.

Generalization of previous techniques to non-isomorphic factors.

Abstract

In previous work, we determined the metric dimension for a direct product of three isomorphic complete graphs. Turning to the case where the complete graphs may have different orders, there are three families we refer to as the upper, lower, and middle cones. We determine the metric dimension and location-total-domination number for a family of direct products of three complete graphs stemming from the middle cone. We explicitly describe minimum resolving sets. To verify the sets are resolving, we define a basic landmark system and show it will be a resolving set if and only if its associated 3-edge-colored hypergraph avoids three types of forbidden subgraphs. This generalizes the technique used for three isomorphic factors.