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Erschienen in:
01.11.2014
On metric dimension of permutation graphs
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 4/2014
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Abstract
The metric dimension
\(\dim (G)\) of a graph \(G\) is the minimum number of vertices such that every vertex of \(G\) is uniquely determined by its vector of distances to the set of chosen vertices. Let \(G_1\) and \(G_2\) be disjoint copies of a graph \(G\), and let \(\sigma : V(G_1) \rightarrow V(G_2)\) be a permutation. Then, a permutation graph
\(G_{\sigma }=(V, E)\) has the vertex set \(V=V(G_1) \cup V(G_2)\) and the edge set \(E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}\). We show that \(2 \le \dim (G_{\sigma }) \le n-1\) for any connected graph \(G\) of order \(n\) at least \(3\). We give examples showing that neither is there a function \(f\) such that \(\dim (G)<f(\dim (G_{\sigma }))\) for all pairs \((G,\sigma )\), nor is there a function \(g\) such that \(g(\dim (G))>\dim (G_{\sigma })\) for all pairs \((G, \sigma )\). Further, we characterize permutation graphs \(G_{\sigma }\) satisfying \(\dim (G_{\sigma })=n-1\) when \(G\) is a complete \(k\)-partite graph, a cycle, or a path on \(n\) vertices.