Opuscula Math. 35, no. 6 (2015), 907-914
http://dx.doi.org/10.7494/OpMath.2015.35.6.907

Opuscula Mathematica

# On vertex stability of complete k-partite graphs

Mateusz Nikodem

Abstract. Let $$H$$ be any graph. We say that graph $$G$$ is $$H$$-stable if $$G-u$$ contains a subgraph isomorphic to $$H$$ for an arbitrary chosen $$u\in V(G)$$. We characterize all $$H$$-stable graphs of minimal size where $$H$$ is any complete $$k$$-partite graph. Thus, we generalize the results of Dudek and Żak regarding complete bipartite graphs.

Keywords: vertex stability, minimal stable graphs, complete $$k$$-partite graphs.

Mathematics Subject Classification: 05C35, 05C60.

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• Mateusz Nikodem
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
• Revised: 2015-01-19.
• Accepted: 2015-03-16.

Mateusz Nikodem, On vertex stability of complete k-partite graphs, Opuscula Math. 35, no. 6 (2015), 907-914, http://dx.doi.org/10.7494/OpMath.2015.35.6.907

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