scholarly journals A NOTE ON EDGE-CONNECTIVITY OF THE CARTESIAN PRODUCT OF GRAPHS

2011 ◽  
Vol 84 (1) ◽  
pp. 171-176
Author(s):  
LAKOA FITINA ◽  
C. T. LENARD ◽  
T. M. MILLS
Keyword(s):  
Cartesian Product ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Cartesian Product Of Graphs ◽  
Necessary And Sufficient ◽  
Product Of Graphs ◽  
Cartesian Products Of Graphs

AbstractThe main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

scholarly journals Strong geodetic problem on Cartesian products of graphs

2018 ◽  
Vol 52 (1) ◽  
pp. 205-216 ◽  
Author(s):  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Cartesian Product Of Graphs ◽  
Geodetic Number ◽  
Recent Variation ◽  
Product Of Graphs ◽  
Strong Geodetic Number ◽  
Cartesian Products Of Graphs

The strong geodetic problem is a recent variation of the geodetic problem. For a graph G, its strong geodetic number sg(G) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on a fixed shortest path between a pair of vertices from S. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for sg(G □ H) is determined, as well as exact values for Km □ Kn, K1,k □ Pl, and prisms over Kn–e. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.

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

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.

scholarly journals On very strongly perfect Cartesian product graphs

2021 ◽  
Vol 7 (2) ◽  
pp. 2634-2645
Author(s):  
Ganesh Gandal ◽  
◽  
R Mary Jeya Jothi ◽  
Narayan Phadatare ◽  
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Perfect Graph ◽  
Finite Graphs ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Necessary And Sufficient ◽  
And Control ◽  
Product Of Graphs ◽  
Real Time Application

<abstract><p>Let $ G_1 \square G_2 $ be the Cartesian product of simple, connected and finite graphs $ G_1 $ and $ G_2 $. We give necessary and sufficient conditions for the Cartesian product of graphs to be very strongly perfect. Further, we introduce and characterize the co-strongly perfect graph. The very strongly perfect graph is implemented in the real-time application of a wireless sensor network to optimize the set of master nodes to communicate and control nodes placed in the field.</p></abstract>

scholarly journals Peg Solitaire on Cartesian Products of Graphs

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.

s-strongly perfect cartesian product of graphs

1992 ◽  
Vol 16 (4) ◽  
pp. 297-303
Author(s):  
Elefterie Olaru ◽  
Eugen M??ndrescu
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
M. R. CHITHRA ◽  
A. VIJAYAKUMAR
Keyword(s):  
Interconnection Networks ◽  
Cartesian Product ◽  
Graph Products ◽  
Single Edge ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

scholarly journals On Topologies Induced by Graphs Under Some Unary and Binary Operations

2019 ◽  
Vol 12 (2) ◽  
pp. 499-505
Author(s):  
Caen Grace Sarona Nianga ◽  
Sergio R. Canoy Jr.
Keyword(s):  
Undirected Graph ◽  
Cartesian Product ◽  
Binary Operations ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

Let G = (V (G),E(G)) be any simple undirected graph. The open hop neighborhood of v ϵ V(G) is the set 𝑁_𝐺^2(𝑣) = {u ϵ V(G):  𝑑_𝐺 (u,v) = 2}. Then G induces a topology τ_G on V (G) with base consisting of sets of the form F_G^2[A] = V(G) \ N_G^2 [A] where N_G^2 [A] = A ∪ {v ϵ V(G):  𝑁_𝐺^2(𝑣) ∩ A ≠ ∅ } and A ranges over all subsets of V (G). In this paper, we describe the topologies induced by the complement of a graph, the join, the corona, the composition and the Cartesian product of graphs.

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