DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2022): 0.7

5-year Journal Impact Factor (2022): 0.7

CiteScore (2022): 1.9

SNIP (2022): 0.902

Discussiones Mathematicae Graph Theory

Article in volume

Authors:

T. Paj Erker

Tjaša Paj Erker

.

email: tjasa.paj@um.si

S. Špacapan

Simon Špacapan

University of Maribor, FME, and IMFM

email: simon.spacapan@um.si

Title:

Separation of Cartesian products of graphs into several connected components by the removal of vertices

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Source:

Discussiones Mathematicae Graph Theory 42(3) (2022) 905-920

Received: 2019-10-17 , Revised: 2020-03-12 , Accepted: 2020-03-13 , Available online: 2020-04-14 , https://doi.org/10.7151/dmgt.2315

Abstract:

A set $S\subseteq V(G)$ 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 $\kappa_k(G)$, is the minimum cardinality of a vertex $k$-cut in $G$. We give several constructions of a set $S$ such that $(G\Box H)-S$ 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 $G\Box H$. This yields a formula for $\kappa_3(G\Box H)$.

Keywords:

$k$-connectivity, Cartesian product

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