Geodesic
The upper and lower bounds for the number of additional arcs in a minimal edge $1$-extension of oriented cycle
Prikladnaya Diskretnaya Matematika. Supplement, no. 15 (2022), pp. 112-116

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A $k$-edge extension of a graph $G$ with $n$ vertices is minimal if it has $n$ vertices and contains the minimum number of edges or arcs among all $k$-edge extensions of $G$ with $n$ vertices. Minimal edge $1$-extensions of cycles are well studied. In this paper, we consider minimal edge $1$-extensions of cycle orientations. We study the upper and lower bounds for the number of additional arcs $\text{ec}(C_n)$ of a minimal edge $1$-extension of the oriented cycle $C_n$. The main result is an estimate for the number of additional arcs: $\left\lceil {n}/{2} \right\rceil \leq \text{ec}(C_n) \leq n$. Examples of cycle orientations on which the upper and lower bounds are achieved are given.
Keywords: minimal edge extension, fault-tolerance.
Mots-clés : cycle orientation 
@article{PDMA_2022_15_a26,
     author = {O. V. Modenova and M. B. Abrosimov},
     title = {The upper and lower bounds for the number of additional arcs in a minimal edge $1$-extension of oriented cycle},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {112--116},
     publisher = {mathdoc},
     number = {15},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2022_15_a26/}
}
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O. V. Modenova; M. B. Abrosimov. The upper and lower bounds for the number of additional arcs in a minimal edge $1$-extension of oriented cycle. Prikladnaya Diskretnaya Matematika. Supplement, no. 15 (2022), pp. 112-116. http://geodesic.mathdoc.fr/item/PDMA_2022_15_a26/