The concertina graph is the 18-vertex, 27-edge cubic Hamiltonian graph illustrated above. It has girth 5, graph diameter 4, and chromatic number 3, and is non-apex.
The name appears to refer to the concertina-like visual appearance of the graph in the embedding presented by Robertson et al. (2019, Fig. 15).
A cubic graph is cyclically 5-connected if it has girth at least 5 and every edge cut separating two subgraphs that contain cycles has at least five edges. It is dodecahedrally-connected if no 5-edge cut having at least seven vertices on each side cuts off a subgraph that can be drawn in a disk with its five attachment vertices on the boundary.
The characterization of apex dodecahedrally-connected graphs states that such a graph is apex if and only if it contains none of the Petersen graph, jaws graph, starfish graph, Robertson diamond graph, concertina graph, and bigdrum graph as a topological minor (Robertson et al. 2019).
The concertina graph is implemented in the Wolfram Language as GraphData["ConcertinaGraph"].
