Optimal 1-edge fault-tolerant designs for ladders

A graph G* is 1-edge fault-tolerant with respect to a graph G, denoted by 1-EFT(G), if every graph obtained by removing any edge from G* contains G. A 1-EFT(G) graph is optimal if it contains the minimum number of edges among all 1-EFT(G) graphs. The kth ladder graph, Lk, is defined to be the cartesian product of the Pk and P2 where Pn is the n-vertex path graph. In this paper, we present several 1-edge fault-tolerant graphs with respect to ladders. Some of these graphs are proven to be optimal.

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