Talk:Good spanning tree

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Orderly Spanning Trees

Latest comment: 9 months ago1 comment1 person in discussion

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:

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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 ${\displaystyle T}$ .

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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