Lately, something I've been interested in is finding the set of forbidden minors for the apex graphs. One thing I tried to do was to look at the graphs which were known, and try to find a pattern in the planar subgraphs that result from deleting two vertices. However, when trying this on the Petersen family, what came out of it didn't look that nice. Luckily, the $Y-\Delta$ transformations give a simple characterization of the Petersen family as the equivalence class containing, say, $K_6$ under $Y-\Delta$ transformations.After some preliminary googling, I came across a blog post by David Eppstein in which he lists a bunch of forbidden minors for apex graphs by carefully attaching copies of $K_5$ and $K_{3,3}$ together. I was hoping to find some more not listed there (besides the trivial disjoint unions of $K_5$ and $K_{3,3}$) using $Y-\Delta$ transformations, but found that sometimes the resulting graphs are apex. Is there any characterization (or just weak conditions) of when apexness is preserved?
Are there any conditions for which ($k$-)apexness is preserved under $Y-\Delta$ transformations?
graph theory;co.combinatorics