The jaws graph is the 20-vertex, 30-edge cubic Hamiltonian graph illustrated above. It has girth 5, graph crossing number 2, graph diameter 5, and chromatic number 3.
A cubic graph is theta-connected if it has girth at least 5, every edge cut separating two subgraphs containing cycles has at least five edges, and every edge cut having at least seven vertices on each side has at least six edges (Robertson et al. 2019).
The jaws graph was defined in a characterization of theta-connected graphs excluding the Petersen graph as a topological minor. Any such graph containing the jaws graph as a topological minor is doublecross. Together with the starfish graph and Petersen graph, the jaws graph is one of the three minimal non-apex obstructions among theta-connected graphs (Robertson et al. 2019).
The jaws graph is implemented in the Wolfram Language as GraphData["JawsGraph"].
