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.

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2022_42_3_a13,
     author = {Erker, Tja\v{s}a Paj and \v{S}pacapan, Simon},
     title = {Separation of {Cartesian} {Products} of {Graphs} {Into} {Several} {Connected} {Components} by the {Removal} of {Vertices}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {905--920},
     publisher = {mathdoc},
     volume = {42},
     number = {3},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2022_42_3_a13/}
}
TY  - JOUR
AU  - Erker, Tjaša Paj
AU  - Špacapan, Simon
TI  - Separation of Cartesian Products of Graphs Into Several Connected Components by the Removal of Vertices
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2022
SP  - 905
EP  - 920
VL  - 42
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2022_42_3_a13/
LA  - en
ID  - DMGT_2022_42_3_a13
ER  - 
%0 Journal Article
%A Erker, Tjaša Paj
%A Špacapan, Simon
%T Separation of Cartesian Products of Graphs Into Several Connected Components by the Removal of Vertices
%J Discussiones Mathematicae. Graph Theory
%D 2022
%P 905-920
%V 42
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2022_42_3_a13/
%G en
%F DMGT_2022_42_3_a13
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/

[1] D. Bauer, H. Broersma and E. Schmeichel, Toughness in graphs—A survey, Graphs Combin. 22 (2006) 1–35. https://doi.org/10.1007/s00373-006-0649-0

[2] G. Chartrand, S.F. Kapoor, L. Lesniak and D.R. Lick, Generalized connectivity in graphs, Bull. Bombay Math. Colloq. 2 (1984) 1–6.

[3] V. Chvátal, Tough graphs and Hamiltonian circuits, Discrete Math. 5 (1973) 215–228. https://doi.org/10.1016/0012-365X(73)90138-6

[4] D.P. Day, O.R. Oellermann and H.C. Swart, The ℓ-connectivity function of trees and complete multipartite graphs, J. Combin. Math. Combin. Comput. 10 (1991) 183–192.

[5] J. Govorčin, and R. Škrekovski, On the connectivity of Cartesian product of graphs, Ars Math. Contemp. 7 (2014) 293–297. https://doi.org/10.26493/1855-3974.313.e10

[6] O.R. Oellermann, On the ℓ-connectivity of a graph, Graphs Combin. 3 (1987) 285–291. https://doi.org/10.1007/BF01788551

[7] G. Sabidussi, Graphs with given group and given graph-theoretical properties, Canad. J. Math. 9 (1957) 515–525. https://doi.org/10.4153/CJM-1957-060-7

[8] Y. Sun and X. Li, On the difference of two generalized connectivities of a graph, J. Comb. Optim. 33 (2017) 283–291. https://doi.org/10.1007/s10878-015-9956-9

[9] Y. Sun, F. Li and Z. Jin, On two generalized connectivities of graphs, Discuss. Math. Graph Theory 38 (2018) 245–261. https://doi.org/10.7151/dmgt.1987

[10] Y. Sun, On the maximum and minimum sizes of a graph with given k-connectivity, Discuss. Math. Graph Theory 37 (2017) 623–632. https://doi.org/10.7151/dmgt.1941

[11] S. Špacapan, Connectivity of Cartesian products of graphs, Appl. Math. Lett. 21 (2008) 682–685. https://doi.org/10.1016/j.aml.2007.06.010

[12] J.M. Xu and C. Yang, Connectivity and super-connectivity of Cartesian product graphs, Ars Combin. 95 (2010) 235–245.

[13] C. Yang and J.M. Xu, Reliability of interconnection networks modeled by Cartesian product digraphs, Networks 52 (2008) 202–205. https://doi.org/10.1002/net.20231