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Connectivity and other invariants of generalized products of graphs
- Source :
- UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC), Repositorio Abierto de la UdL, Universitad de Lleida, Recercat. Dipósit de la Recerca de Catalunya, Universitat Jaume I
- Publication Year :
- 2015
-
Abstract
- Figueroa-Centeno et al. [4] introduced the following product of digraphs let D be a digraph and let Γ be a family of digraphs such that V (F) = V for every F∈Γ . Consider any function h:E(D)→Γ . Then the product D⊗hΓ is the digraph with vertex set V(D)×V and ((a,x),(b,y))∈E(D⊗hΓ) if and only if (a,b)∈E(D) and (x,y)∈E(h(a,b)) . In this paper, we deal with the undirected version of the ⊗h -product, which is a generalization of the classical direct product of graphs and, motivated by the ⊗h -product, we also recover a generalization of the classical lexicographic product of graphs, namely the ∘h -product, that was introduced by Sabidussi in 1961. We provide two characterizations for the connectivity of G⊗hΓ that generalize the existing one for the direct product. For G∘hΓ , we provide exact formulas for the connectivity and the edge-connectivity, under the assumption that V (F) = V , for all F∈Γ . We also introduce some miscellaneous results about other invariants in terms of the factors of both, the ⊗h -product and the ∘h -product. Some of them are easily obtained from the corresponding product of two graphs, but many others generalize the existing ones for the direct and the lexicographic product, respectively. We end up the paper by presenting some structural properties. An interesting result in this direction is a characterization for the existence of a nontrivial decomposition of a given graph G in terms of ⊗h -product. The research conducted in this document by the first author has been supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387.
- Subjects :
- ¿h-product
direct product
Grafs, Teoria de
General Mathematics
Lexicographic product of graphs
Matemàtiques i estadística [Àrees temàtiques de la UPC]
Vertex (geometry)
Graph theory
Algebra
Combinatorics
Computer Science::Discrete Mathematics
Product (mathematics)
connectivity
¿h
Direct product
lexicographic product
Teoria de grafs
Mathematics
Directed graphs
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC), Repositorio Abierto de la UdL, Universitad de Lleida, Recercat. Dipósit de la Recerca de Catalunya, Universitat Jaume I
- Accession number :
- edsair.doi.dedup.....a06293131bb55bd93ad1c708dc09d55c