The superbox graph is the 16-vertex, 24-edge cubic Hamiltonian graph illustrated above. It has girth 5, graph crossing number 2, graph diameter 4, and chromatic number 3. It is also an apex graph. In the notation of Robertson et al. (2019), it is isomorphic to the graph obtained from the box graph by performing a graph subdivision on each of two edges and joining the new vertices.
The name "superbox" reflects this construction as an extension of the box graph.
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.
The superbox lemma states that if a cyclically 5-connected graph containing the superbox graph contains neither the Petersen graph nor the triplex graph as a topological minor, and the distinguished edge of the superbox is mapped to a single edge, then deleting that edge makes the containing graph planar (Robertson et al. 2019).
The superbox graph is implemented in the Wolfram Language as GraphData["SuperboxGraph"].
