Good spanning tree

In the mathematical field of graph theory, a good spanning tree $$T$$ of an embedded planar graph $$G$$ is a rooted spanning tree of $$G$$ whose non-tree edges satisfy the following conditions.
 * there is no non-tree edge $$(u,v)$$ where $$u$$ and $$v$$ lie on a path from the root of $$T$$ to a leaf,
 * the edges incident to a vertex $$v$$ can be divided by three sets $$X_v, Y_v $$ and $$ Z_v$$, where,
 * $$X_v$$ is a set of non-tree edges, they terminate in red zone
 * $$Y_v$$ is a set of tree edges, they are children of $$v$$
 * $$Z_v$$ is a set of non-tree edges, they terminate in green zone

Formal definition
Source:

Let $$G_\phi$$ be a plane graph. Let $$T$$ be a rooted spanning tree of $$G_\phi$$. Let $$P(r,v)=(r=u_1), u_2, \ldots, (v=u_k)$$ be the path in $$T$$ from the root $$r$$ to a vertex $$v\ne r$$. The path $$P(r,v)$$ divides the children of $$u_i$$, $$(1\le i < k)$$, except $$u_{i+1}$$, into two groups; the left group $$L$$ and the right group $$R$$. A child $$x$$ of $$u_i$$ is in group $$L$$ and denoted by $$u_{i}^L$$ if the edge $$(u_i,x)$$ appears before the edge $$(u_i, u_{i+1})$$ in clockwise ordering of the edges incident to $$u_i$$ when the ordering is started from the edge $$(u_i,u_{i+1})$$. Similarly, a child $$x$$ of $$u_i$$ is in the group $$R$$ and denoted by $$u_{i}^R$$ if the edge $$(u_i,x)$$ appears after the edge $$(u_i, u_{i+1})$$ in clockwise order of the edges incident to $$u_i$$ when the ordering is started from the edge $$(u_i,u_{i+1})$$. The tree $$T$$ is called a  good spanning tree  of $$G_\phi$$ if every vertex $$v$$ $$(v\ne r)$$ of $$G_\phi$$ satisfies the following two conditions with respect to $$P(r,v)$$.


 * [Cond1] $$G_\phi$$ does not have a non-tree edge $$(v,u_i)$$, $$i<k$$; and
 * [Cond2] the edges of $$G_\phi$$ incident to the vertex $$v$$ excluding $$(u_{k-1},v)$$ can be partitioned into three disjoint (possibly empty) sets $$X_v,Y_v$$ and $$Z_v$$ satisfying the following conditions (a)-(c)
 * (a) Each of $$X_v$$ and $$Z_v$$ is a set of consecutive non-tree edges and $$Y_v$$ is a set of consecutive tree edges.
 * (b) Edges of set $$X_v$$, $$Y_v$$ and $$Z_v$$ appear clockwise in this order from the edge $$(u_{k-1}, v)$$.
 * (c) For each edge $$(v,v')\in X_v$$, $$v'$$ is contained in $$T_{u_i^L}$$, $$i<k$$, and for each edge $$(v,v')\in Z_v$$, $$v'$$ is contained in $$T_{u_i^R}$$, $$i<k$$.Good spanning tree example.svg

Applications
In monotone drawing of graphs, in 2-visibility representation of graphs.

Finding good spanning tree
Every planar graph $$G$$ has an embedding $$G_\phi$$ such that $$G_\phi$$ contains a good spanning tree. A good spanning tree and a suitable embedding can be found from $$G$$ in linear-time. Not all embeddings of $$G$$ contain a good spanning tree.