The strong metric dimension of graphs and digraphs

Authors:
Ortrud R. Oellermann;Joel Peters-Fransen
Affiliations:
The University of Winnipeg, 515 Portage Avenue, Winnipeg, MB, Canada R3B 2E9;The University of Winnipeg, 515 Portage Avenue, Winnipeg, MB, Canada R3B 2E9
Venue:
Discrete Applied Mathematics
Year:
2007

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Abstract

Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u,v of vertices of G if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong dimension of G. It is shown that the problem of finding the strong dimension of a connected graph can be transformed to the problem of finding the vertex covering number of a graph. Moreover, it is shown that computing this invariant is NP-hard. Related invariants for directed graphs are defined and studied.