On the edge-connectivity and restricted edge-connectivity of a product of graphs

Abstract

The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters ( and , edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G m ,

The p-restricted edge-connectivity of Kneser graphs

“…It has been shown that λ p ( G ) ≤ ξ p ( G ) for many graphs [4,6,16,21,28,30] and sufficient conditions to establish that λ p (G ) = ξ p (G ) have been given in [4,18,26] among others.…”

Separation of Cartesian products of graphs into several connected components by the removal of edges

“…The connectivity of graph products was also studied in [1] and [2] where the authors determine connectivity properties of the * -product, and use these properties to construct networks with high fault-tolerance. The * -product was first defined in [3] and is a concept which is more general than the Cartesian product of graphs (i.e.…”

Vulnerability of super extra edge-connected graphs

“…Fàbrega and Fiol [9,10] generalized the concept of connectivity and edge-connectivity proposed by Harary [11], to k-extra connectivity and k-extra edge-connectivity. [1,5,6,13,17,18,20,21,27,34]. Hsieh and Chang [17] discussed the 3-extra connectivity of a k-ary n-cube for k ≥ 4 and n ≥ 5.…”