Geodesic
On the number of optimal $1$-hamiltonian graphs with the number of vertices up to~$26$ and~$28$
Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 103-105

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A graph is called $1$-vertex-hamiltonian ($1$-edge-hamiltonian) one, if it becomes Hamiltonian after deleting any its vertex (edge). $1$-vertex-hamiltonian ($1$-edge-hamilton) graph is optimal if it has the minimum number of edges among all $1$-vertex-hamiltonian ($1$-edge-hamiltonian) graphs with the same number of vertices. In the paper, the previous data on the number of optimal $1$-vertex- and $1$-edge-hamiltonian graphs with the number of vertices up to $26$ are verified, and new data for $28$-vertex graphs are given.
Keywords: optimal $1$-hamiltonian graph, minimal $1$-extension of cycle, fault-tolerance. 
@article{PDMA_2016_9_a39,
     author = {M. B. Abrosimov and S. A. Suhov},
     title = {On the number of optimal $1$-hamiltonian graphs with the number of vertices up to~$26$ and~$28$},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {103--105},
     publisher = {mathdoc},
     number = {9},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2016_9_a39/}
}
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M. B. Abrosimov; S. A. Suhov. On the number of optimal $1$-hamiltonian graphs with the number of vertices up to~$26$ and~$28$. Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 103-105. http://geodesic.mathdoc.fr/item/PDMA_2016_9_a39/