Geodesic
On a~connection between the switching separability of a~graph and of its subgraphs
Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 2, pp. 46-56.

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A graph of order $n\geq4$ is called switching separable if the modulo-2 sum with some complete bipartite graph on the same vertex set results in a graph consisting of two mutually independent subgraphs of orders at least two. We prove that if removal of one or two vertices of the graph always results in a switching-separable subgraph, then the graph itself is switching separable. On the other hand, for every odd order there exists a nonswitching-separable graph such that removal of any one vertex gives a switching-separable subgraph. We also show connections with similar facts for the separability of Boolean functions and $n$-ary quasigroups. Ill. 1, bibl. 6.
Keywords: graph connectivity, graph switching, $n$-ary quasigroups, reducibility, Seidel switching, separability, two-graphs. 
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D. S. Krotov. On a~connection between the switching separability of a~graph and of its subgraphs. Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 2, pp. 46-56. http://geodesic.mathdoc.fr/item/DA_2010_17_2_a3/

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[2] Krotov D. S., “On reducibility of $n$-ary quasigroups”, Discrete Math., 308:22 (2008), 5289–5297 | DOI | MR | Zbl

[3] Krotov D. S., Potapov V. N., Sokolova P. V., “On reconstructing reducible $n$-ary quasigroups and switching subquasigroups”, Quasigroups Relat. Syst., 16:1 (2008), 55–67 | MR | Zbl

[4] Krotov D. S., Potapov V. N., “$n$-Ary quasigroups of order 4”, SIAM J. Discrete Math., 23:2 (2009), 561–570 | DOI | MR

[5] Krotov D. S., Potapov V. N., On connection between reducibility of an $n$-ary quasigroup and that of its retracts, Electronic preprint, Submitted, arXiv: 0801.0055

[6] Spence E., “Two-graphs”, CRC Handbook of combinatorial designs, CRC Press, Boca Raton, FL, 1996, 686–694 | Zbl