Good spanning tree

Conditions of good spanning tree

 an illustration for $X_{v},Y_{v}$ and $Z_{v}$ sets of edges

In the mathematical field of graph theory, a good spanning tree T of an plane graph G is a rooted spanning tree of G whose non-tree edges satisfy the following condition.

• there is no non-tree edge $(u,v)$ where $u$ and $v$ lie on a path from root to a leaf.
• the edges incident to a vertex v can be divided by three set of $X_{v},Y_{v},Z_{v}$
  • $X_{v}$ is a set of non-tree edges, each of them termites in red zone
  • $Y_{v}$ is a set of tree edges, each of them is a child of $v$
  • $Z_{v}$ is a set of non-tree edges, each of them termites in green zone

Formal Definition

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 $v\neq r$. The path $P(r,v)$ divides the children of $u_{i}$, $(1\leq i<k)$, 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 $(u_{i},u_{i+1})$. We call $T$ a {\it good spanning} tree of $G_{\phi }$ if every vertex $v$ $(v\neq r)$ of $G$ satisfies the following two conditions with respect to $P(r,v)$.

• [(Cond1)] $G$ does not have a non-tree edge $(v,u_{i})$, $i<k$; and

A plane graph G (left), a good spanning tree of G (right) solid edges are part of spanning tree and dotted edges are non-tree edges

• [(Cond2)] the edges of $G$ 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 $T_{u_{i}^{R}}$, $i<k$.

Applications

in monotone drawing

See also

• Spanning tree
• Schnyder realizer