On the reliability of generalized Petersen graphs

The diameter vulnerability of the generalized Petersen graphGP[tk;k]

“…Thus, for t odd, t ≥ 3 , P (3) is the shortest path from u 0 to u N provided that k ≥ 3 or t ≥ 5 when k = 2 , and we have…”

“…For k to divide N , then ⌊ k 2 ⌋ = 0 , which is not possible since k ≥ 2 . Thus k does not divide N and (a) considering the path P (3) from u 0 to u N , we get |E(P (3)…”

“…For k = 1 , it is easy to observe that the deletion of any edge e from G = GP [t, 1] leaves the diameter of the graph G ′ 1 = G − e equal to that of G except when t = 3 and e is a spoke, in which case D(GP [3,1] − e) = 3 instead of 2. Thus, in the sequel we will focus on values of k ≥ 2 .…”

“…Thus, |E(P (1) )| = |E(P (2) )| = k − i + 3 (and hence P (1) and P (2) can be used interchangeably) and |E(P (3) )| = i + 6 . We note that |E(P (1) )| ≤ |E(P (3) )| if and only if i ≥ (1) )| = |E(P (2)…”

“…However, since We remark that from the way that P (1) , P (2) and P (3) were defined, and recalling that P (1) and P (2) can be used interchangeably, we get that…”