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| Content Provider | Springer Nature Link |
|---|---|
| Author | Yi, Eunjeong |
| Copyright Year | 2013 |
| Abstract | A vertex x in a graph G strongly resolves a pair of vertices v,w if there exists a shortest x − w path containing v or a shortest x − v path containing w in G. A set of vertices S ⊆ V (G) is a strong resolving set of G if every pair of distinct vertices of G is strongly resolved by some vertex in S. The strong metric dimension of G, denoted by sdim(G), is the minimum cardinality over all strong resolving sets of G. For a connected graph G of order n ≥ 2, we characterize G such that sdim(G) equals 1, n − 1, or n − 2, respectively. We give a Nordhaus-Gaddum-type result for the strong metric dimension of a graph and its complement: for a graph G and its complement $$\bar G$$ , each of order n ≥ 4 and connected, we show that $$2 \leqslant sdim\left( G \right) + sdim\left( {\bar G} \right) \leqslant 2\left( {n - 2} \right)$$ . It is readily seen that $$sdim\left( G \right) + sdim\left( {\bar G} \right) = 2$$ if and only if n = 4; we show that, when G is a tree or a unicyclic graph, $$sdim\left( G \right) + sdim\left( {\bar G} \right) = 2\left( {n - 2} \right)$$ if and only if n = 5 and $$G \cong \bar G \cong C_5$$ , the cycle on five vertices. For connected graphs G and $$\bar G$$ of order n ≥ 5, we show that $$3 \leqslant sdim\left( G \right) + sdim\left( {\bar G} \right) \leqslant 2\left( {n - 3} \right)$$ if G is a tree; we also show that $$4 \leqslant sdim\left( G \right) + sdim\left( {\bar G} \right) \leqslant 2\left( {n - 3} \right)$$ if G is a unicyclic graph of order n ≥ 6. Furthermore, we characterize graphs G satisfying $$sdim\left( G \right) + sdim\left( {\bar G} \right) = 2\left( {n - 3} \right)$$ when G is a tree or a unicyclic graph. |
| Ending Page | 1492 |
| Page Count | 14 |
| Starting Page | 1479 |
| File Format | |
| ISSN | 14398516 |
| e-ISSN | 14397617 |
| Journal | Acta Mathematica Sinica, English Series |
| Issue Number | 8 |
| Volume Number | 29 |
| Language | English |
| Publisher | Springer Berlin Heidelberg |
| Publisher Date | 2013-07-15 |
| Publisher Place | Berlin/Heidelberg |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Trees unicyclic graph strong metric dimension Strong resolving set Nordhaus-Gaddum-type Distance in graphs tree Mathematics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics |
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