On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
Sakander Hayat, Asad Khan, Yubin Zhong
Abstract
Graphs of order n with fault-tolerant metric dimension n have recently been characterized.This paper points out an error in the proof of this characterization. We show that the complete multipartite graphs also have the fault-tolerant metric dimension n, which provides an infinite family of counterexamples to the characterization. Furthermore, we find exact values of the metric, edge metric, mixed-metric dimensions, the domination number, locating-dominating number, and metric-locating-dominating number for the complete multipartite graphs. These results generalize various results in the literature from complete bipartite to complete multipartite graphs.
Topics & Concepts
MultipartiteMathematicsMetric dimensionCounterexampleCombinatoricsMetric (unit)Bipartite graphDiscrete mathematicsDimension (graph theory)Chordal graph1-planar graphGraphPhysicsEconomicsQuantumQuantum mechanicsQuantum entanglementOperations managementGraph Labeling and Dimension Problemsgraph theory and CDMA systemsAdvanced Graph Theory Research