In this paper we determine when the join of two graphs is locally homogeneous. It is shown that the Cartesian product of a locally H1 graph and a locally H2 graph is locally (H1∪ H2). All graphs H of order at most 4 for which there are no locally H graphs are determined.

We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of interval-valued fuzzy complete graphs and present some properties of self complementary and self weak complementary interval-valued fuzzy complete graphs.

The Cartesian product G x H of graphs G and H has as vertices the pairs (g, h) with g a vertex of G and h a vertex of H; (gl, hI) is connected by an edge to (g2' h2) in G x H just when {gl' g2} is an edge of G and hI = h2' or when g, = g2 and {h" h2} is an edge of H. The Cartesian product admits unique factorization (Sabidussi [4]) but until recently no efficient algorithm was known for produci...

Let G H denote the Cartesian product of the graphs G and H. In 2004, Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24(3) (2004), 389–402] characterized prism fixers, i.e., graphs G for which γ(G K2) = γ(G), and noted that γ(G Kn) ≥ min{|V (G)|, γ(G)+n−2}. We call a graph G a consistent fixer if γ(G Kn) = γ(G)+n−2 for each n such that 2 ≤ n...

Using a newly introduced operation on graphs and its counterpart on graph drawings, we prove the conjecture of Jendrol’ and Ščerbová from 1982 about the crossing number of the Cartesian product K1,m2Pn. Our approach is applicable to the capped Cartesian products of Pn with any graph containing a dominating vertex.