Euler s formula n e + f = 2 and Platonic solids

Transcription

1 Euler s formula n e + f = and Platonic solids

2 Euler s formula n e + f = and Platonic solids spherical projection makes these planar graphs

3 Euler s formula n e + f = and Platonic solids spherical projection makes these planar graphs p = vertex degree, q = face degree pn = e qf = e Euler implies e p e + e q = so p + q > 1 Thus (p, q) {(, ), (, 4), (4, ), (, ), (, )}

4 Planar graph Planar drawing

5 Planar graph Planar (straight-line) drawing

6 Planar drawing Planar Embedding :,14,10 : 1,,7 :,11,4 4:,,6 : 4,8,1,7,6 6: 4, 7:, 8:,9,1 9: 8,11,10 10: 1,9,11 11:,10,9 1: 8,14,1 1:,1,14 14: 1,1,1 Outer face: 1,14,1,,7, (Clockwise order of neighbors.)

7 Kuratowski s Theorem G is planar iff it doesn t contain a subgraph that is a subdivision of K or K,.

8 Kuratowski s Theorem G is planar iff it doesn t contain a subgraph that is a subdivision of K or K,.

9 Planar graph DFS orientation & numbering tree edge back edge

10 DFS orientation & numbering LR partition [Brandes 11]

11 How to order outgoing edges? Two (related) mechanisms: Nesting (tree edges) 10 (6,8) cycle must be nested inside (6,7) cycle Left / Right Partitioning (back edges) both (7,) and (11,) can t bend the same way

12 How to order outgoing edges? Two (related) mechanisms: Nesting (tree edges) 10 (6,8) cycle must be nested inside (6,7) cycle Left / Right Partitioning (back edges) both (7,) and (11,) can t bend the same way

13 The return point of a back edge (v, w) is w. The return points of a tree edge (v, w) are u such that u + v w x u return point of (6, 4) is 4. return point of (, 7) is. return points of (4, ) are,,1.

14 The return point of a back edge (v, w) is w. The return points of a tree edge (v, w) are u such that u + v w x u. The lowpt of an edge (v, w) is its lowest return point (or w if none exists). lowpt of (6, 4) is 4. lowpt of (, 7) is. lowpt of (4, ) is 1.

15 The return point of a back edge (v, w) is w. The return points of a tree edge (v, w) are u such that u + v w x u. The lowpt of an edge (v, w) is its lowest return point (or w if none exists). Back edge (x, u) is the return edge for itself and every tree edge (v, w) with u + v w x u. 1

16 14 11 An LR partition is a partition of the back edges into Left and Right so 1 10 that for every fork e 1 v e 1 9 e all return edges of e 1 ending strictly higher than lowpt(e ) belong to one partition, and 4 all return edges of e ending strictly higher than lowpt(e 1 ) belong to the other. 1

17 14 11 An LR partition is a partition of the back edges into Left and Right so 1 10 that for every fork e 1 v e 1 9 e all return edges of e 1 ending strictly higher than lowpt(e ) belong to one partition, and 4 all return edges of e ending strictly higher than lowpt(e 1 ) belong to the other. 1 Fork at v = 1 implies (14, 1) and (1, ) must be in different LR partitions

18 e 1 v e e all return edges of e 1 ending strictly higher than lowpt(e ) belong to one partition lowpt(e ) lowpt(e 1 ) all return edges of e ending strictly higher than lowpt(e 1 ) belong to the other. same constraint = all...belong different constraint = belong to the other e 1 v e e lowpt(e ) lowpt(e 1 )

19 Theorem. A graph is planar if and only if it has an LR partition (based on an arbitrary DFS orientation). Proof. Use LR-partition to create a nesting order on outgoing edges at each vertex. (Assign tree edge to same LR-partition as its return edge with the highest return point.) e 1 v e e e 1 v e e lowpt(e ) e 1 e lowpt(e 1 ) lowpt(e ) lowpt(e 1 )

20 Let e L 1 e L l be the left outgoing edges and e R 1 e R r be the right outgoing edges at v then the (edge) embedding at v is: (u, v), L(e L l ), el l, R(eL l ),..., L(eL 1 ), e L 1, R(e L 1 ), R(e R 1 ), e R 1, R(e R 1 ),..., L(e R r ), e R r, R(e R r ) R(e L l ) e L l e L 1 e R 1 v L(e L u l ) e R r where L(e) and R(e) denote the left and right back edges to v whose cycles share e. Within R(e) (and L(e)), back edges are ordered using (and ) applied to the fork of their cycles.

21 Algorithm 1) For every pair of back edges b 1 and b, determine if they should be in the same or different LR-partitions. e 1 e different iff lowpt(e ) < lowpt(b 1 ) and lowpt(e 1 ) < lowpt(b ) b b 1

22 Algorithm 1) For every pair of back edges b 1 and b, determine if they should be in the same or different LR-partitions. e 1 w b 1 x e b different iff lowpt(e ) < lowpt(b 1 ) and lowpt(e 1 ) < lowpt(b ) same if lowpt((x, w)) < min{lowpt(b 1 ), lowpt(b )} for some (x, w) where x is shared by C(b 1 ) and C(b ) but w is not.

23 Algorithm ) Create a constraint graph on back edges: constraint edge (b 1, b ) is red if b 1 and b are different, or blue if the same. ) Find a balanced bipartition of the constraint graph: red edges connect vertices (i.e., back edges) in different partitions, blue edges connect vertices in the same partition.