Connectivity and other invariants of generalized products of graphs

TL;DR

This paper introduces an undirected generalization of a graph product based on a digraph product, explores its connectivity and invariants, and relates it to classical graph products and intersection graphs.

Contribution

It defines a new undirected graph product generalizing the classical direct and lexicographic products, and studies its connectivity and invariants.

Findings

Introduces the undirected $igotimes_h$-product of graphs.

Establishes relationships between invariants of the product and factors.

Identifies a new intersection graph related to connectivity.

Abstract

Figueroa-Centeno et al. introduced the following product of digraphs: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D)\longrightarrow\Gamma $. Then the product $D\otimes_{h} \Gamma$ is the digraph with vertex set $V(D)\times V$ and $((a,x),(b,y))\in E(D\otimes_h\Gamma)$ if and only if $(a,b)\in E(D)$ and $(x,y)\in E(h (a,b))$. In this paper, we introduce the undirected version of the $\otimes_h$-product, which is a generalization of the classical direct product of graphs and, motivated by it, we also recover a generalization of the classical lexicographic product of graphs that was introduced by Sabidussi en 1961. We study connectivity properties and other invariants in terms of the factors. We also present a new intersection graph that emerges when we characterize the connectivity of $\otimes_h$-product of graphs.