Geodesic
On conditional connectivity of the Cartesian product of cycles
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 17-34

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The conditional h-vertex (h-edge) connectivity of a connected graph H of minimum degree k gt;h is the size of a smallest vertex (edge) set F of H such that H - F is a disconnected graph of minimum degree at least h. Let G be the Cartesian product of r≥ 1 cycles, each of length at least four and let h be an integer such that 0≤ h≤ 2r-2. In this paper, we determine the conditional h-vertex-connectivity and the conditional h-edge-connectivity of the graph G. We prove that both these connectivities are equal to (2r-h)a_h^r, where a_h^r is the number of vertices of a smallest h-regular subgraph of G.
Keywords: fault tolerance, hypercube, conditional connectivity, cut, Cartesian product 
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Saraf, J.B.; Borse, Y.M.; Mundhe, Ganesh. On conditional connectivity of the Cartesian product of cycles. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 17-34. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a1/