ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

#### For citation:

Abrosimov M. B., Kamil Iehab A. K., Lobov A. A. Construction of All Nonisomorphic Minimal Vertex Extensions of the Graph by the Method of Canonical Representatives. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 4, pp. 479-486. DOI: 10.18500/1816-9791-2019-19-4-479-486, EDN: YXOMDX

Published online:
02.12.2019
Full text:
Language:
Russian
Article type:
Article
UDC:
519.17
EDN:
YXOMDX

# Construction of All Nonisomorphic Minimal Vertex Extensions of the Graph by the Method of Canonical Representatives

Autors:
Abrosimov Mikhail Borisovich, Saratov State University
Kamil Iehab A. K., Saratov State University
Lobov Alexandr A., Saratov State University
Abstract:

In 1976 John P. Hayes proposed a graph model for investigating the fault tolerance of discrete systems. The technical system is mapped to a graph. The elements of the system correspond to the vertices of the graph, and links between the elements correspond to edges or arcs of the graph. Failure of a system element refers to the removal of the corresponding vertex from the system graph along with all its edges. Later together with Frank Harary the model was extended to links failures. The formalization of a fault-tolerant system implementation is the extension of the graph. The graph G* is called the vertex k-extension of the graph G if after removing any k vertices from the graph G* result graph contains the graph G. A vertex k-extension of the graph G is called minimal if it has the least number of vertices and edges among all vertex k-extensions of the graph G. An algorithm for constructing all nonisomorphic minimal vertex k-extensions of the given graph using the method of canonical representatives is proposed.

Key words:
References:
1. Hayes J. P. A graph model for fault-tolerant computing system. IEEE Transactions on Computers, 1976, vol. C-25, iss. 9, pp. 875–884. DOI: https://doi.org/10.1109/TC.1976.1674712
2. Harary F., Hayes J. P. Edge fault tolerance in graphs. Networks, 1993, vol. 23. pp. 135– 142. DOI: https://doi.org/10.1002/net.3230230207
3. Abrosimov M. B. Grafovye modeli otkazoustoichivosti [Fault tolerance graph models]. Saratov, Izd-vo Sarat. un-ta, 2012. 192 p. (in Russian).
4. Bogomolov A. M., Salii V. N. Algebraicheskie osnovy teorii diskretnykh sistem [Algebraic foundations of the theory of discrete systems]. Moscow, Nauka, 1997. 368 p. (in Russian).
5. Abrosimov M. B. On the complexity of some problems related to graph extensions. Math. Notes, 2010, vol. 88, iss. 5, pp. 619–625. DOI: https://doi.org/10.1134/S0001434610110015
6. Abrosimov M. B. Minimal graph extensions. In: Novye informatsionnye tekhnologii v issledovanii diskretnykh struktur [New Information Technologies in the Study of Discrete Structures]. Tomsk, 2000, pp. 59–64 (in Russian).
7. Abrosimov M. B. Minimal’nye rasshireniia 4-, 5-, 6- i 7-vershinnykh grafov [Minimal extension of graphs with 4, 5, 6 and 7 vertices]. Saratov, Saratov State University, 2000, 26 p.; VINITI 06.09.2000, no. 2352-В00 (in Russian).
8. Brinkmann G. Isomorphism rejection in structure generation programs. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 2000, vol. 51, pp. 25–38. DOI: https://doi.org/10.1090/dimacs/051/03
9. McKay B. D., Piperno A. Practical Graph Isomorphism, II. Journal of Symbolic Computation, 2014, vol. 60, pp. 94–112. DOI: https://doi.org/10.1016/j.jsc.2013.09.003
10. Volga Regional Center for New Information Technologies. Available at: http://prcnit.sgu.ru (accessed 1 May 2019) (in Russian).
11. Graph World. Available at: http://graphworld.ru (accessed 1 May 2019) (in Russian).