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 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 Number in Some Networks

2019 ◽  
Vol 11 (2) ◽  
pp. 20
Author(s):  
Huifen Ge ◽  
Zhao Wang ◽  
Jinyu Zou
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Geodetic Number ◽  
Product Graphs ◽  
Geodetic Set ◽  
Cartesian Product Graphs ◽  
Strong Geodetic Number

A vertex subset S of a graph is called a strong geodetic set if there exists a choice of exactly one geodesic for each pair of vertices of S in such a way that these (|S| 2) geodesics cover all the vertices of graph G. The strong geodetic number of G, denoted by sg(G), is the smallest cardinality of a strong geodetic set. In this paper, we give an upper bound of strong geodetic number of the Cartesian product graphs and study this parameter for some Cartesian product networks.

scholarly journals On the Gonality of Cartesian Products of Graphs

10.37236/9307 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Ivan Aidun ◽  
Ralph Morrison
Keyword(s):  
Upper Bound ◽  
Algebraic Curve ◽  
Cartesian Product ◽  
Metric Graphs ◽  
Systematic Treatment ◽  
Cartesian Products ◽  
Product Graphs ◽  
Chip Firing ◽  
Cartesian Products Of Graphs

In this paper we provide the first systematic treatment of Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve defined in terms of chip-firing.  We prove an upper bound on the gonality of the Cartesian product of any two graphs, and determine instances where this bound holds with equality, including for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$.  We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound.  We also extend some of our results to metric graphs.

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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