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universal fixer; domination
Given two disjoint copies of a graph $G$, denoted $G^1$ and $G^2$, and a permutation $\pi $ of $V(G)$, the graph $\pi G$ is constructed by joining $u \in V(G^1)$ to $\pi (u) \in V(G^2)$ for all $u \in V(G^1)$. $G$ is said to be a universal fixer if the domination number of $\pi G$ is equal to the domination number of $G$ for all $\pi $ of $V(G)$. In 1999 it was conjectured that the only universal fixers are the edgeless graphs. Since then, a few partial results have been shown. In this paper, we prove the conjecture completely.
[1] Burger, A. P., Mynhardt, C. M.: Regular graphs are not universal fixers. Discrete Math. 310 (2010), 364-368. DOI 10.1016/j.disc.2008.09.016 | MR 2563053 | Zbl 1216.05098
[2] Cockayne, E. J., Gibson, R. G., Mynhardt, C. M.: Claw-free graphs are not universal fixers. Discrete Math. 309 (2009), 128-133. DOI 10.1016/j.disc.2007.12.053 | MR 2475005 | Zbl 1219.05116
[3] Gibson, R. G.: Bipartite graphs are not universal fixers. Discrete Math. 308 (2008), 5937-5943. DOI 10.1016/j.disc.2007.11.006 | MR 2464884 | Zbl 1181.05068
[4] Gu, W.: Communication with S. T. Hedetniemi. Southeastern Conference on Combinatorics, Graph Theory, and Computing. Newfoundland, Canada, 1999.
[5] Gu, W., Wash, K.: Bounds on the domination number of permutation graphs. J. Interconnection Networks 10 (2009), 205-217. DOI 10.1142/S0219265909002522
[6] Hartnell, B. L., Rall, D. F.: On dominating the Cartesian product of a graph and $K_2$. Discuss. Math., Graph Theory 24 (2004), 389-402. DOI 10.7151/dmgt.1238 | MR 2120063 | Zbl 1063.05107
[7] Mynhardt, C. M., Xu, Z.: Domination in prisms of graphs: universal fixers. Util. Math. 78 (2009), 185-201. MR 2499846 | Zbl 1284.05199
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