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

The Maximum Genus of Cartesian Products of Graphs

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-096-2 ◽

1974 ◽

Vol 26 (5) ◽

pp. 1025-1035 ◽

Cited By ~ 11

Author(s):

Joseph Zaks

Keyword(s):

Connected Graph ◽

Connected Components ◽

Open Disk ◽

Maximum Genus ◽

Cartesian Products ◽

Euler’S Formula ◽

Cartesian Products Of Graphs

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formulaV-E + F = 2-2g.

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

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm160719018s ◽

2021 ◽

pp. 18-18

Author(s):

Simon Spacapan

Keyword(s):

Cartesian Product ◽

Connected Components ◽

Edge Connectivity ◽

Cartesian Products ◽

Minimum Cardinality ◽

Connected Graphs ◽

Cartesian Products Of Graphs

Let G = (V (G),E(G)) be a graph. A set S ? E(G) is an edge k-cut in G if the graph G-S = (V (G), E(G) \ S) has at least k connected components. The generalized k-edge connectivity of a graph G, denoted as ?k(G), is the minimum cardinality of an edge k-cut in G. In this article we determine generalized 3-edge connectivity of Cartesian product of connected graphs G and H and describe the structure of any minimum edge 3-cut in G2H. The generalized 3-edge connectivity ?3(G2H) is given in terms of ?3(G) and ?3(H) and in terms of other invariants of factors G and H.

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Peg Solitaire on Cartesian Products of Graphs

Graphs and Combinatorics ◽

10.1007/s00373-021-02289-7 ◽

2021 ◽

Vol 37 (3) ◽

pp. 907-917

Author(s):

Martin Kreh ◽

Jan-Hendrik de Wiljes

Keyword(s):

Cartesian Product ◽

Cartesian Products ◽

Connected Graphs ◽

Grid Graphs ◽

Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

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