The Greenfield graph is the name given in this work to the 12-vertex graph illustrated above due to Greenfield (2011). This graph provided the first known example which answered in the negative the question posed by Gibbons and Laison (2009) asking if the greedy algorithm for computing fixing number is well-defined. For the case of the Greenfield graph, this algorithm can return either 3 or 4, whereas the actual fixing number is 3.
Greenfield (2011, p. 44) proposed that this graph is a minimal counterexample. E. Weisstein (Jun. 6, 2023) found no smaller counterexample in search up to 10 nodes using a particular choice strategy for ties, but did identify a small number of larger ones.
The Greenfield graph is implemented in the Wolfram Language as GraphData["GreenfieldGraph"].
