The window graph is the 12-vertex, 18-edge cubic Hamiltonian graph illustrated above. It has girth 4, graph crossing number 1, graph diameter 3, and chromatic number 3. It is also an apex graph.
The name appears to refer to the window-like visual appearance of the nested rectangular frames in the embedding presented by Robertson et al. (2019, Fig. 9).
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 characterization states that a cyclically 5-connected cubic graph containing the window graph but containing none of the Petersen graph, triplex graph, box graph, antibox graph, and drape graph as a topological minor has graph crossing number at most 1 (Robertson et al. 2019).
The window graph is implemented in the Wolfram Language as GraphData["WindowGraph"].
