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As far as I can see, the definition of Good spanning trees is equivalent to the definition of Orderly Spanning Trees, which were introduced by Yi-Ting Chiang, Ching-Chi Lin and Hsueh-I Lu at SODA 2001: https://epubs.siam.org/doi/10.1137/S0097539702411381 / https://arxiv.org/abs/cs/0102006

This is their definition:

Let ${\displaystyle T}$ be a rooted spanning tree of a connected plane graph ${\displaystyle H}$. Two distinct nodes of ${\displaystyle H}$ are unrelated with respect to ${\displaystyle T}$ if neither of them is an ancestor of the other in ${\displaystyle T}$. An edge ${\displaystyle e}$ of ${\displaystyle H}$ is unrelated with respect to ${\displaystyle T}$ if the endpoints of ${\displaystyle e}$ are unrelated with respect to ${\displaystyle T}$. Let ${\displaystyle v_{1},v_{2},\dots ,v_{n}}$ be the counterclockwise preordering of the nodes in ${\displaystyle T}$. A node ${\displaystyle v_{i}}$ is orderly in ${\displaystyle H}$ with respect to ${\displaystyle T}$ if the neighbors of ${\displaystyle v_{i}}$ in ${\displaystyle H}$ form the following four blocks of ${\displaystyle H}$ with respect to ${\displaystyle T}$ in counterclockwise order around ${\displaystyle v_{i}}$:

• ${\displaystyle B_{1}(v_{i})}$: the parent of ${\displaystyle v_{i}}$ in ${\displaystyle T}$;
• ${\displaystyle B_{2}(v_{i})}$: the nodes ${\displaystyle v_{j}}$ with ${\displaystyle j<i}$ that are unrelated to ${\displaystyle v_{i}}$ with respect to ${\displaystyle T}$;
• ${\displaystyle B_{3}(v_{i})}$: the children of ${\displaystyle v_{i}}$ in ${\displaystyle T}$; and
• ${\displaystyle B_{4}(v_{i})}$: the nodes ${\displaystyle v_{j}}$ with ${\displaystyle j>i}$ that are unrelated to ${\displaystyle v_{i}}$ with respect to ${\displaystyle T}$, where each block could be empty.

${\displaystyle T}$ is an orderly spanning tree of ${\displaystyle H}$ if (i) ${\displaystyle v_{1}}$ is on the contour of ${\displaystyle H}$, and (ii) each node ${\displaystyle v_{i}}$ is orderly in ${\displaystyle H}$ with respect to ${\displaystyle T}$.

In terms of the definition of a good spanning tree, it seems to me that

• ${\displaystyle B_{1}(v)}$ is just the parent of ${\displaystyle v}$,
• ${\displaystyle B_{2}(v)}$ is equivalent to ${\displaystyle Z_{v}}$,
• ${\displaystyle B_{3}(v)}$ is equivalent to ${\displaystyle Y_{v}}$, and
• ${\displaystyle B_{4}(v)}$ is equivalent to ${\displaystyle X_{v}}$.

So I think that the definitions are equivalent. Can someone please check whether I'm correct? PhKindermann (talk) 13:19, 25 November 2024 (UTC)Reply