Geodesic
On minimal vertex 1-extensions of special type graph union
Prikladnaâ diskretnaâ matematika, no. 4 (2011), pp. 34-41

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In 2001, it was conjectured that the minimal vertex 1-extension of a graph $G+G^*$, where $G^*$ is a minimal vertex 1-extension of graph $G$, is unique up to isomorphism and has the form $G^*+G^*$. We construct two counterexamples to this conjecture showing that, in general, it is wrong. Also, we show that the statement is true for many graphs.
Keywords: graph, minimal vertex extension, exact vertex extension, fault tolerance. 
@article{PDM_2011_4_a4,
     author = {M. B. Abrosimov},
     title = {On minimal vertex 1-extensions of special type graph union},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {34--41},
     publisher = {mathdoc},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2011_4_a4/}
}
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M. B. Abrosimov. On minimal vertex 1-extensions of special type graph union. Prikladnaâ diskretnaâ matematika, no. 4 (2011), pp. 34-41. http://geodesic.mathdoc.fr/item/PDM_2011_4_a4/