The Tilley graph is a polyhedral graph that arises in the investigation of Kempe chainsKempe Chain (Tilley 2018). It has 12 vertices, 29 edges, and 19 faces (one of which is a quadrilateral and the remainder of which are triangular). It can be constructed by removing any single edge from the icosahedral graph (E. Weisstein, Sep. 9, 2022).
There are 6 symmetrically distinct triangular faces out of the 18 total triangular faces.
It is implemented in the Wolfram Language as GraphData["TilleyGraph"].
