Vulnerability of super edge-connected networks

• Authors: Zhen-Mu Hong;Jun-Ming Xu
• Venue: Theoretical Computer Science
• Year: 2014

When the underlying topology of an interconnection network is modeled by a connected graph G, the connectivity of G is an important measurement for reliability and fault tolerance of the network. For a given integer h=0, a subset F of edges in a connected graph G is an h-extra edge-cut if G-F is disconnected and every component has more than h vertices. The h-extra edge-connectivity @l^(^h^)(G) of G is defined as the minimum cardinality over all h-extra edge-cuts of G. A graph G, if @l^(^h^)(G) exists, is super-@l^(^h^) if every minimum h-extra edge-cut of G isolates at least one connected subgraph of order h+1. The persistence @r^(^h^)(G) of a super-@l^(^h^) graph G is the maximum integer m for which G-F is still super-@l^(^h^) for any set F@?E(G) with |F|@?m. Hong et al. (2012) [12] showed that min{@l^(^1^)(G)-@d(G)-1,@d(G)-1}@?@r^(^0^)(G)@?@d(G)-1, where @d(G) is the minimum vertex-degree of G. This paper shows that min{@l^(^2^)(G)-@x(G)-1,@d(G)-1}@?@r^(^1^)(G)@?@d(G)-1, where @x(G) is the minimum edge-degree of G. In particular, for a k-regular super-@l^(^1^) graph G, @r^(^1^)(G)=k-1 if @l^(^2^)(G) does not exist or G is super-@l^(^2^) and triangle-free, from which the exact values of @r^(^1^)(G) are determined for some well-known networks.