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How Much Work Does It Take To Straighten a Plane Graph Out?M.Kang ∗M.Schacht †O.Verbitsky ‡Abstract We prove that if one wants to make a plane graph drawing straight-line then in the worst ca one has to move almost all vertices.We u the standard concepts of a plane graph and a plane embedding (or drawing )of an abstract planar graph (,[1]).Given a plane graph G ,we want to redraw it making all its edges straight line gments while keeping as many vertices on the spot as possible.Let shift (G )denote the smallest s such that we can do the job by shifting only s vertices.We define s (n )to be the maximum shift (G )over all G with n vertices.The function s (n )can have another interpretation cloly related to a nice web puzzle called Planarity Game [3].At the start of the game,a player es a straight line drawing of a planar graph with many edge crossings.In a move s/he is allowed to shift one vertex to a new position;the incident edges are redrawn correspondingly (being all the time straight line gments).The objective is to obtain a crossing-free
drawing.Thus,s (n )is equal to the number of moves that the player,playing optimally,is forced to make on an n -vertex game instance at the worst ca.
The Wagner-F´a ry-Stein theorem (,[2])says that every G has a straight line plane embedding and immediately implies an upper bound s (n )≤n −3.We here aim at proving a lower bound.
Given an abstract planar graph G ,let shift (G )denote the maximum shift (G ′)over all plane embeddings G ′of G .Thus,we are eking for G with
英文报纸>不要介意 英文
large shift(G).Every4-connected planar graph G is Hamiltonian(Tutte[4]), therefore,has a matching of size at least(n−1)/2and,therefore,shift(G)≥(n−3)/2.An example of planar G with3k vertices and shift(G)≥2k−8 is shown in[5],thereby giving us a bound s(n)>2
1There is no need to describe edges;we can suppo either that the drawing is straight line with edge crossings as in the Planarity Game or that we have an arbitrary crossing-free drawing with edg
es of any shape.
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plane version of G.That is,either the two embeddings are obtainable from one another by a plane homeomorphism or this is true after changing outer face in one of them.By construction,the regions occupied by the G[V i]’s in the original embedding are pairwi disjoint.If we change outer face, this is still true possibly with one exception.It follows that all but one T i’s are pairwi disjoint.Without loss of generality suppo that the possible exception is T s+k.
Call V′i persistent if i<s+k and|V′i∩V i|≥2.Since all persistent T i’s are pairwi disjoint and each of them contains a pair of vertices of some regular k-gon,there can be at most k−1persistent ts.It follows that the number of moved vertices is at least
s+k−1
i=1|V′i\V i|≥s(k−1)=(
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k
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)n−k2+k,
日文原来如此as claimed.
