An Apex graph is a grpah that is either Planar, or can be made planar by removing a single vertex. Are all Apex graphs Toroidal?
2026-03-27 01:43:09.1774575789
Are Apex graphs always Toroidal?
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According to this paper (Figure 7, first example), the graph consisting of two copies of $K_5$ with one identified vertex is not toroidal, but it is clearly apex by removing the identified vertex since $K_4$ is planar. So the answer is no.