We investigate graphs and hypergraphs that have (relaxed) product structures. In the class of graphs, we discuss in detail RSP-relations, a relaxation of relations fulfilling the square property and therefore of the product relation σ, that identifies the copies of the prime factors of a graph w.r.t. the Cartesian product. For K2,3-free graphs finest RSP-relations can be computed in polynomial-...

We consider classes of graphs that enjoy the following properties: they are closed for gated subgraphs, gated amalgamation and Cartesian products, and for any gated subgraph the inverse of the gate function maps vertices to gated subsets. We prove that any graph of such a class contains a peripheral subgraph which is a Cartesian product of two graphs: a gated subgraph of the graph and a prime g...

We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of Kn × G is found, provided that G has a largest greedy independent decomposition of length c < n. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, an...

The generalized k-connectivity κk(G) of a graphG, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let G and H be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized 3-connectivity of the Cartesian product graph G H and proposed a conjecture for the case that H is 3-connected. In this paper, we give two...

a connected graph g is said to be neighbourly irregular graph if no two adjacent vertices of g have same degree. in this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in at least one interval between the vertices of S. The size of a minimum geodetic set in G is the geodetic number of G. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products hav...

The irregularity of a simple undirected graph G was defined by Albertson [5] as irr(G) = ∑ uv∈E(G) |dG(u)− dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and symmetric difference. ...