Geodesic
Separation of Cartesian Products of Graphs Into Several Connected Components by the Removal of Vertices
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 905-920

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A set S ⊆ V (G) is a vertex k-cut in a graph G = (V (G), E(G)) if G − S has at least k connected components. The k-connectivity of G, denoted as κk(G), is the minimum cardinality of a vertex k-cut in G. We give several constructions of a set S such that (G□H) − S has at least three connected components. Then we prove that for any 2-connected graphs G and H, of order at least six, one of the defined sets S is a minimum vertex 3-cut in G□H. This yields a formula for κ3(G□H).
Keywords: k -connectivity, Cartesian product 
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Erker, Tjaša Paj; Špacapan, Simon. Separation of Cartesian Products of Graphs Into Several Connected Components by the Removal of Vertices. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 905-920. http://geodesic.mathdoc.fr/item/DMGT_2022_42_3_a13/