Connectivity and other invariants of generalized products of graphs

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 $h:E(D)⟶\Gamma$. 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.

arXiv comment14 pages

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