Minimum 3-geodetically connected graphs. (English) Zbl 1029.05045
A graph \(G\) is \(k\)-geodetically connected if it is connected and the removal of at least \(k\) vertices is required to increase the distance between at least one pair of vertices or reduce \(G\) to a single vertex. The author characterizes the class of minimum 3-geodetically connected graphs that have the fewest edges for a given number of vertices.
Reviewer: Alexander Rappoport (Moskva)
MSC:
05C12 | Distance in graphs |
05C35 | Extremal problems in graph theory |
05C38 | Paths and cycles |
05C40 | Connectivity |
05C75 | Structural characterization of families of graphs |
References:
[1] | J.-M. Chang, C.-W, Ho, C.C. Hsu, Y.L. Wang, The characterizations of hinge-free networks, Proceedings of the International Computer Symposium on Algorithms, Taiwan, 1996, pp. 105-112.; J.-M. Chang, C.-W, Ho, C.C. Hsu, Y.L. Wang, The characterizations of hinge-free networks, Proceedings of the International Computer Symposium on Algorithms, Taiwan, 1996, pp. 105-112. |
[2] | Entringer, R. E.; Jackson, D. E.; Slater, P. J., Geodetic connectivity of graphs, IEEE Trans. Circuits and Systems, CAS-24, 460-463 (1977) · Zbl 0344.05137 |
[3] | Farley, A. M.; Proskurowski, A., Minimum self-repairing graphs, Graphs Combin., 13, 345-351 (1997) · Zbl 0890.05059 |
[4] | Harary, F., Graph Theory (1969), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0797.05064 |
[5] | Jackson, D. E.; Entringer, R. E., Minimum \(k\)-geodetically connected graphs, Congr. Numer., 36, 303-309 (1982) · Zbl 0515.05039 |
[6] | Plesnı́k, J., Towards minimum \(k\)-geodetically connected graphs, Networks, 41, 73-82 (2003) · Zbl 1014.05022 |
[7] | J. Plesnı́k, Minimum \(k\); J. Plesnı́k, Minimum \(k\) |
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