Extended split graphs and the 3-sphere regular cellulation conjecture

The 3-sphere regular cellulation conjecture claims that every 2-connected cyclic graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere. The conjecture is obviously true for planar graphs. 2-connectivity is a necessary condition for a graph to satisfy such property. Therefore, the question whether a graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere would be equivalent to the 2-connectivity test if the conjecture were proved to be true. On the contrary, it is not even clear whether such decision problem is computationally tractable. We introduced a new class of graphs called weakly-split and proved the conjecture for such class. Hamiltonian, split, complete k-partite and matrogenic cyclic graphs are weakly split. In this paper, we introduce another class of graphs for which the conjecture is true. Such class is a superclass of planar graphs and weakly-split graphs.