# Distance-Locally Disconnected Graphs

Mirka Miller; Joe Ryan; Zdeněk Ryjáček

Discussiones Mathematicae Graph Theory (2013)

- Volume: 33, Issue: 1, page 203-215
- ISSN: 2083-5892

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topMirka Miller, Joe Ryan, and Zdeněk Ryjáček. "Distance-Locally Disconnected Graphs." Discussiones Mathematicae Graph Theory 33.1 (2013): 203-215. <http://eudml.org/doc/268130>.

@article{MirkaMiller2013,

abstract = {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.},

author = {Mirka Miller, Joe Ryan, Zdeněk Ryjáček},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {neighborhood; distance; locally disconnected; cage; neighborhood distance},

language = {eng},

number = {1},

pages = {203-215},

title = {Distance-Locally Disconnected Graphs},

url = {http://eudml.org/doc/268130},

volume = {33},

year = {2013},

}

TY - JOUR

AU - Mirka Miller

AU - Joe Ryan

AU - Zdeněk Ryjáček

TI - Distance-Locally Disconnected Graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2013

VL - 33

IS - 1

SP - 203

EP - 215

AB - 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.

LA - eng

KW - neighborhood; distance; locally disconnected; cage; neighborhood distance

UR - http://eudml.org/doc/268130

ER -

## References

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- [3] F. Lazebnik, V.A. Ustimenko and A.J. Woldar, New upper bounds on the order of cages, Electron. J. Combin. 4(2) (1977) R13. Zbl0885.05078
- [4] L. Lovász, J. Pelikán and K. Vesztergombi, Discrete Mathematics: Elementary and Beyond (Springer, NewYork, 2003). Zbl1017.00002
- [5] Z. Ryjáček, N2-locally disconnected graphs, Discrete Math. 121 (1993) 189-193. doi:10.1016/0012-365X(93)90551-4[Crossref]
- [6] Z. Ryjáček and B. Zelinka, Locally disconnected graphs with large numbers of edges, Math. Slovaca 37(2) (1987) 195-198.
- [7] B. Zelinka, Two local properties of graphs, ˇ Casop. Pˇest. Mat. 113 (1988) 113-121.

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