Distance-Locally Disconnected Graphs
For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic...
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2013-03-01
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| Series: | Discussiones Mathematicae Graph Theory |
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| Online Access: | https://doi.org/10.7151/dmgt.1657 |
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doaj-a0f7f5976bee479ebe7ff97efa9c47c22021-09-05T17:20:19ZengSciendoDiscussiones Mathematicae Graph Theory2083-58922013-03-0133120321510.7151/dmgt.1657Distance-Locally Disconnected GraphsMiller Mirka0Ryan Joe1Ryjáček Zdeněk2University of Newcastle, Australia University of West Bohemia, Pilsen, Czech Republic King’s College, London, UK ITB Bandung, IndonesiaUniversity of Newcastle, AustraliaUniversity of West Bohemia, Pilsen, Czech Republic Institute for Theoretical Computer Science, Pilsen, Czech Republic University of Newcastle, AustraliaFor an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.https://doi.org/10.7151/dmgt.1657neighborhooddistancelocally disconnectedcage |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Miller Mirka Ryan Joe Ryjáček Zdeněk |
| spellingShingle |
Miller Mirka Ryan Joe Ryjáček Zdeněk Distance-Locally Disconnected Graphs Discussiones Mathematicae Graph Theory neighborhood distance locally disconnected cage |
| author_facet |
Miller Mirka Ryan Joe Ryjáček Zdeněk |
| author_sort |
Miller Mirka |
| title |
Distance-Locally Disconnected Graphs |
| title_short |
Distance-Locally Disconnected Graphs |
| title_full |
Distance-Locally Disconnected Graphs |
| title_fullStr |
Distance-Locally Disconnected Graphs |
| title_full_unstemmed |
Distance-Locally Disconnected Graphs |
| title_sort |
distance-locally disconnected graphs |
| publisher |
Sciendo |
| series |
Discussiones Mathematicae Graph Theory |
| issn |
2083-5892 |
| publishDate |
2013-03-01 |
| description |
For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages. |
| topic |
neighborhood distance locally disconnected cage |
| url |
https://doi.org/10.7151/dmgt.1657 |
| work_keys_str_mv |
AT millermirka distancelocallydisconnectedgraphs AT ryanjoe distancelocallydisconnectedgraphs AT ryjacekzdenek distancelocallydisconnectedgraphs |
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1717786589032087552 |