I'm not quite sure that I understand how we can construct corresponding delta-matroid for a given embedded graph. The text below is from the article "Delta-matroids and Vassiliev invariants" by S. Lando and V. Zhukov [arXiv:1602.00027]:
Firstly, it seems to me that we may choose any path-connected set of edges to be a feasible set $\phi \in \mbox{Ф}$, but then there is a remark that for a plane graph $\phi$ is necessarily a spanning tree.
So, for example, suppose I have the following graph with numerated edges:
My question is may sets of edges $\{1\}$, $\{1,2\}$, $\{1,3\}$ and so on be treated as feasible sets or is there the only possibility to choose a set $\phi$ as $\{1,2,3,4\}$ since this graph has unique spanning tree that coincides with the graph itself?