# Edge-connectivity of strong products of graphs

• Volume: 27, Issue: 2, page 333-343
• ISSN: 2083-5892

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## Abstract

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either ${x}_{i}={y}_{i}$ or ${x}_{i}{y}_{i}\in E\left({G}_{i}\right)$. In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

## How to cite

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Bostjan Bresar, and Simon Spacapan. "Edge-connectivity of strong products of graphs." Discussiones Mathematicae Graph Theory 27.2 (2007): 333-343. <http://eudml.org/doc/270713>.

@article{BostjanBresar2007,
abstract = {The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_i = y_i$ or $x_i y_i ∈ E(G_i)$. In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.},
author = {Bostjan Bresar, Simon Spacapan},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {connectivity; strong product; graph product; separating set; separating sets},
language = {eng},
number = {2},
pages = {333-343},
title = {Edge-connectivity of strong products of graphs},
url = {http://eudml.org/doc/270713},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Bostjan Bresar
AU - Simon Spacapan
TI - Edge-connectivity of strong products of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2007
VL - 27
IS - 2
SP - 333
EP - 343
AB - The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_i = y_i$ or $x_i y_i ∈ E(G_i)$. In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.
LA - eng
KW - connectivity; strong product; graph product; separating set; separating sets
UR - http://eudml.org/doc/270713
ER -

## References

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1. [1] W. Imrich and S. Klavžar, Product graphs: Structure and Recognition (John Wiley & Sons, New York, 2000). Zbl0963.05002
2. [2] S. Spacapan, Connectivity of Cartesian product of graphs, submitted 2005. Zbl1152.05340
3. [3] S. Spacapan, Connectivity of strong products of graphs, submitted 2006. Zbl1213.05154
4. [4] J.M. Xu and C. Yang, Connectivity of Cartesian product graphs, Discrete Math. 306 (2006) 159-165, doi: 10.1016/j.disc.2005.11.010. Zbl1085.05042

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