Fault tolerance of metric basis can be expensive
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2024
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| _version_ | 1866913532245704704 |
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| author | Knor, Martin Sedlar, Jelena Škrekovski, Riste |
| author_facet | Knor, Martin Sedlar, Jelena Škrekovski, Riste |
| contents | A set of vertices S is a resolving set of a graph G; if for every pair of vertices x and y in G, there exists a vertex s in S such that x and y differ in distance to s. A smallest resolving set of G is called a metric basis. The metric dimension dim(G) is the cardinality of a metric basis of G. The notion of a metric basis is applied to the problem of placing sensors in a network, where the problem of sensor faults can arise. The fault-tolerant metric dimension ftdim(G) is the cardinality of a smallest resolving set S such that S\{s} remains a resolving set of G for every s in S. A natural question is how much more sensors need to be used to achieve a fault-tolerant metric basis. It is known in literature that there exists an upper bound on ftdim(G) which is exponential in terms of dim(G); i.e. ftdim(G) <= dim(G)(1+2^(5dim(G)-1)). In this paper, we construct graphs G with ftdim(G) = dim(G)+2^(dim(G)-1) for any value of dim(G), so the exponential upper bound is necessary. We also extend these results to the k-metric dimension which is a generalization of the fault-tolerant metric dimension. First, we establish a similar exponential upper bound on dim(k+1)(G) in terms of dim(k)(G); and then we show that there exists a graph for which dim(k+1)(G) is indeed exponential. For a possible further work, we leave the gap between the bounds to be reduced. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_03656 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fault tolerance of metric basis can be expensive Knor, Martin Sedlar, Jelena Škrekovski, Riste Combinatorics 05C12 A set of vertices S is a resolving set of a graph G; if for every pair of vertices x and y in G, there exists a vertex s in S such that x and y differ in distance to s. A smallest resolving set of G is called a metric basis. The metric dimension dim(G) is the cardinality of a metric basis of G. The notion of a metric basis is applied to the problem of placing sensors in a network, where the problem of sensor faults can arise. The fault-tolerant metric dimension ftdim(G) is the cardinality of a smallest resolving set S such that S\{s} remains a resolving set of G for every s in S. A natural question is how much more sensors need to be used to achieve a fault-tolerant metric basis. It is known in literature that there exists an upper bound on ftdim(G) which is exponential in terms of dim(G); i.e. ftdim(G) <= dim(G)(1+2^(5dim(G)-1)). In this paper, we construct graphs G with ftdim(G) = dim(G)+2^(dim(G)-1) for any value of dim(G), so the exponential upper bound is necessary. We also extend these results to the k-metric dimension which is a generalization of the fault-tolerant metric dimension. First, we establish a similar exponential upper bound on dim(k+1)(G) in terms of dim(k)(G); and then we show that there exists a graph for which dim(k+1)(G) is indeed exponential. For a possible further work, we leave the gap between the bounds to be reduced. |
| title | Fault tolerance of metric basis can be expensive |
| topic | Combinatorics 05C12 |
| url | https://arxiv.org/abs/2410.03656 |