Tutte Embeddings of Planar Graphs

Transcription

1 Spectral Graph Theory and its Applications Lectre 21 Ttte Embeddings o Planar Graphs Lectrer: Daniel A. Spielman November 30, Ttte s Theorem We sally think o graphs as being speciied by vertices and edges. Bt, planar graphs have another ndamental nit: aces. Formally, a ace in a planar graph is a minimal simple cycle o edges. Ttte proved that i one takes a 3-connected planar graph, ixes the vertices o one ace so that the vertices are the corners o a convex polygon, and the edges the edges, and then let every other vertex be the average o its neighbors, then one obtains a planar straight-line embedding o the graph. The only property that we will se o 3-connected graphs in the proo is that or every 5 vertices, v 0, v 1, s, t there are either vertex-disjoint paths rom v 0 to s and v 1 to t that do not go throgh, or vertex-disjoint paths rom v 0 to t and v 1 to s that do not go throgh. I ll present a proo o Ttte s theorem rom the paper One-Forms on Meshes and Applications to 3D Mesh Parameterization by Gortler, Gotsman and Thrston (Harvard CS TR-12-04). The only act abot Ttte embeddings that we will se in the proo is that every non-bondary vertex is a strict convex combination o its neighbors. That is, is can be written as a sm o its neighbors with non-zero coeicients Planarity Given some planar embedding o a planar graph, we can deine an orientation o the edges o every ace. By dealt, we will assme that the edges o all internal aces are oriented clockwise, and the external ace is oriented conter-clockwise. Whenever we consider an edge at a ace, we will consider it with its orientation. A Ttte embedding assigns to each vertex coordinates (x(), y()). We let (G, x, y) denote a Ttte embedding. For a ace in the graph, we say that the embedding o is convex i the polygon corresponding to the ace is convex. That is, i v 1,..., v k are the vertices on the ace in order, then no horizontal line crosses more than two o the edges (v i 1, v i ), where we take v 0 = v k. 21-1

2 Lectre 21: November 30, We say that the ace is strictly convex i in addition we have that 1. no edge has zero length, 2. the angle at each vertex lies strictly between 0 and π, and 3. the ace has non-zero area. (note that the third is implied by the irst and second). For a vertex, let v 1,..., v k denote its neighbors in order. We can then consider the angles in the embedding between v i 1 and v i at. We denote these angles α 1,..., α k. As is a convex combination o its neighbors, we have i α i 2π. We say that is a wheel i i α i = 2π. We say that is a strict wheel i in addition each α i is strictly between 0 and π. We will make se o the ollowing proposition, whose proo I leave as an exercise. Proposition I (G, x, y) is a Ttte embedding o a planar graph in which every internal ace is strictly convex and every internal vertex is a strict wheel, then it is a planar straight-line embedding o G. Or proo that Ttte embeddings are planar will ollow rom the ollowing two theorems. Theorem In a Ttte embedding o a planar graph, every ace is convex and every vertex is a wheel. Theorem In a Ttte embedding o a three-connected planar graph, there are no edges o zero length, no angles o 0 or π and no aces o zero volme Potentials We will begin by establishing the notation we need to prove Theorem Or irst step will be to deine a potential to be a map rom the vertex set o the graph to the reals: σ : V IR. We say that a potential is valid i or all edges (, v), σ() σ(v). We say that two edges (, v 0 ) and (, v 1 ) are a corner i, v 0 and v 1 all lie on some ace, with between v 0 and v 1 in the orientation o the ace. Given a potential, we deine a change to be a corner (, v 0 ), (, v 1 ) sch that 1. some ace contains both edges with orientation (v 0, ) and (, v 1 ), and 2. sign(σ() σ(v 0 )) sign(σ() σ(v 1 )). Note that a change can be associated with both the ace and the vertex. For each vertex, we deine changes() = the nmber o changes at

3 Lectre 21: November 30, and or each ace, we deine changes() = the nmber o changes at. Note that to conte the changes at we mst be carel to consider the orientations. Proposition For every valid potential, changes() = changes(). At aces, we will be more interested in the nmber o non-changes, deined by non changes() = the nmber o sides o changes(). Proposition For every valid potential on a planar graph changes() + non changes() = 2(V 2) + F, where V is the nmber o vertices and F is the nmber o aces in the graph. Proo. We have changes() + non changes() = changes() + (sides() changes()) = sides() = 2E, where E is the nmber o edges in the graph. The proposition now ollows rom Eler s Theorem, which says V E + F = 2. Given a potential, we will say that a vertex is bracketed by its neighbors i it has some neighbors v i and v j or which σ(v i ) σ() σ(v j ). The key to the proo o Theorem is: Lemma Let σ be a valid potential on a planar graph in which there are niqe vertices o maximm and minimm potential, and every other vertex is bracketed by its neighbors. Then, 1. For every non-extreme vertex, changes() = 2. and 2. For every ace, non changes() = 2.

4 Lectre 21: November 30, Proo. As each o the non-extreme vertices are bracketed by their neighbors, each o them mst have at least two changes. Similarly, or each ace, each local minimm and maximm o the potential mst contribte a non-change. So, each ace mst have at least two non-changes. Now, i each non-extreme vertex has the minimal possible nmber o changes, 2, each extreme vertex has no changes, and each ace has the minimal possible nmber o non-changes, 2, then we get changes() + non changes() = 2(V 2) + F. As this agrees with Proposition , each o these vales mst be at their minimm. For all bt a inite nmber o lines throgh the origin, the projection o a Ttte embedding onto that line is a potential satisying the conditions o Lemma Proo o Theorem Let (G, x, y) be a Ttte embedding, and let be a ace. I is not convex, then there is a continos amily o lines that ct in at least 4 edges. Ths, we can ind a line that cts in at least 4 edges sch that the projection o the embedding onto is a valid potential. Moreover, becase each vertex in a Ttte embedding is a convex combination o its neighbors, each non-extreme vertex is bracketed by its neighbors. However, between each consective pair o edges that are ct by the line, mst contain a local maximm or minimm. So, mst contain at least 4 non-changes, contradicting Lemma Similarly, i is not a wheel, then there mst be some line throgh intersecting at least or wedges o the orm (v i 1,, v i ) sch that the projection o the embedding onto is valid and bracketed. However, each o these wedges will contribte a change, contradicting Lemma Proo o Theorem Given a potential σ, we will call an edge (, v) degenerate i σ() = σ(v). We will call a corner (, v 0 ), (, v 1 ) degenerate i both its edges are degenerate. Finally, we will call a vertex degenerate i each o its edges is degenerate. In this section, we will prove Theorem Let (G, x, y) be a Ttte embedding o a 3-connected planar graph, and let σ be a potential obtained by projecting the embedding onto a line. Then, G has no degenerate corner nder σ. Beore proving this theorem, let s observe that it implies Theorem B.

5 Lectre 21: November 30, First, i there is an angle o 0 or π at a corner, (, v 0 ), (, v 1 ), then by projecting onto a line orthogonal to (, v 0 ), we obtain a potential nder which this corner is degenerate. To see that is implies there is no edge o zero length, let (, v 0 ) be sch an edge, and let (, v 1 ) be another edge that together with the irst orms a corner. Now, project the embedding onto a line perpendiclar to the line connecting to v 1. Then, the corner will be degenerate nder this embedding. To prove Theorem , we will irst prove that a degerate corner can only happen at a degernerate vertex. Lemma Let (G, x, y) be a Ttte embedding o a 3-connected planar graph, and let σ be a potential obtained by projecting the embedding onto a line. I (, v 0 ), (, v 1 ) is a degenerate corner, then mst be degenerate. Proo. We irst note that i were non-degenerate, then it wold have some neighbors v i and v j sch that v i < < v j. Let s be one o the vertices at which σ is maximized and let t be one o the vertices at which σ is minimized. Note that both o these mst lie on bondary ace. As the graph is three connected, there are either vertex-disjoint paths rom v 0 to s and v 1 to t that do not go throgh, or vertex-disjoint paths rom v 0 to t and v 1 to s that do not go throgh. Let s assme withot loss o generality that we are in the irst case. Then, there is a simple path in the graph rom t to v 1 to to v 0 to s. It is not too diiclt to show that there exists a valid bracketed potential τ sch that τ(w) τ(v) or all vertices w v, τ(t) = 1, τ(s) = 1, and τ is monotone strictly decreasing on this path. By now considering the potentials σ + ɛτ and σ ɛτ, or siciently small ɛ, we can show that 1. or one o these potentials, has at least 4 changes, and 2. or siciently small ɛ, both o these potentials are bracketed. Ths, we obtain a contradiction to Lemma Finally, we observe that Lemma can be sed to show that i there is any degenerate corner, then every interior vertex is degenerate. To prove this, consider a degenerate vertex, and two o

6 Lectre 21: November 30, its neighbors v 0 and v 1. I there is no edge between v 0 and v 1, add one. The graph remains planar, 3-connected, and the embedding remains a Ttte embedding since v 0 and v 1 mapped to the same point anyway. Now, v 0 has a degenerate corner (v 0, ), (v 0, v 1 ), so v 0 mst be degernate, and so on.