Tardif, C., Prefibers and the Cartesian product of metric spaces, Discrete Mathematics 109 (1992) 283-288. The properties of certain sets called prefibers in a metric space are used to show that the algebraic properties of the Cartesian product of graphs generalize to metric spaces.

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspect...

Given a graph G, a dominating set D is a set of vertices such that any vertex not in D has at least one neighbor in D. A {k}-dominating multiset Dk is a multiset of vertices such that any vertex in G has at least k vertices from its closed neighborhood in Dk when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) to prove a ‘‘Vizing-like’’ inequ...

Let G be an undirected graph with n vertices. Assume that a robot is placed on a vertex and n − 2 obstacles are placed on the other vertices. A vertex on which neither a robot nor an obstacle is placed is said to have a hole. Consider a single player game in which a robot or obstacle can be moved to adjacent vertex if it has a hole. The objective is to take the robot to a fixed destination vert...

let g be a simple connected graph. the generalized polarity wiener index ofg is defined as the number of unordered pairs of vertices of g whosedistance is k. some formulas are obtained for computing the generalizedpolarity wiener index of the cartesian product and the tensor product ofgraphs in this article.

In this paper, the weighted Szeged indices of Cartesian product and Corona product of two connected graphs are obtained. Using the results obtained here, the weighted Szeged indices of the hypercube of dimension n, Hamming graph, C4 nanotubes, nanotorus, grid, t−fold bristled, sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.

We present some results on the growth in various products of graphs. In particular we study the Cartesian, strong, lexicographic, tensor and free product of graphs. We show that with respect to distances the tensor product behaves differently from other products. In general the results are valid for rooted graphs but have especially nice structure in the case of vertex-transitive factors.

Let P = G H be the cartesian product of graphs G,H. We relate the cover time COV[P ] of P to the cover times of its factors. When one of the factors is in some sense larger than the other, its cover time dominates, and can become of the same order as the cover time of the product as a whole. Our main theorem effectively gives conditions for when this holds. The probabilistic technique which we ...

Graphs which can be represented as nontrivial subgraphs of Cartesian product graphs are characterized. As a corollary it is shown that any bipartite, K2,3-free graph of radius 2 has such a representation. An infinite family of graphs which have no such representation and contain no proper representable subgraph is also constructed. Only a finite number of such graphs have been previously known.

The (k − 1)-fault diameter Dk(G) of a k-connected graph G is the maximum diameter of G− F for any F ⊂ V (G) with |F | < k. Krishnamoorthy and Krishnamurthy first proposed this concept and gave Dκ(G1)+κ(G2)(G1 ×G2) Dκ(G1)(G1)+ Dκ(G2)(G2) when κ(G1 ×G2)= κ(G1)+ κ(G2), where κ(G) is the connectivity of G. This paper gives a counterexample to this bound and establishes Dk1+k2(G1 × G2) Dk1(G1) +Dk2 ...