Graph Theory

Ch6 Planar Graphs

Basic Definitions curve, polygon curve, drawing crossing, planar, planar embedding,

and plane graph open set region, face

Proposition: K5 and K3,3 is not planar

Restricted Jordan Curve Theorem

Theorem. A simple closed polygonal curve C consisting of finite number of segments partitions the plane into exactly 2 faces, each have C as boundary

Dual Graphs Definition:

let function F on a graph F(G) = { faces of G }if there exits f: V(G*) → F(G) so that f is 1-1 & onto and for all x, y V(G*), there is an edge connects x, y iff there is an edge e in G that f(x) and f(y) are on the different side of e.

A cut-edge in G becomes a loop in G*

For all x V(G*) and X = f(x), x is in the interior of X Each edge e in G there is exactly one e* in G* that e and e* cro

sses. (G*) *=G iff G is connected

pf:a) for all G, G* is connectedb) each face in G* contains exactly one vertex of G

Two embeddings of a planar graph may have non-isomorphic duals.

Length of a face

length of a face is defined as total length of the boundary of the face.

2e(G) = ∑L (Fi) Theorem.

edges in G form a cycle in G iff the corresponding edges in G* form a bond in G*

Theorem.the follows are equivalentA) G is bipartiteB) every face of G has even lengthC) G* is Eulerian

Outerplane graph Def: outerplanar, outerplane graph

a graph is outerplanar if it has an embedding that every vertex is on the boundary of the unbounded face.

The boundary of the outer face of a 2-connected outerplane graph is a spanning cycle

K4 and K2,3 are not outerplanar.

Every simple outerplane graph has 2 non-adjacent vertex of degree at most 2pf:1. n(G) < 3, every vertex has degree ≤22. n(G) = 4 holds. (think about K4 – {any edges})3. n(G) ≥ 4

Euler’s Formula n – e + f = 2

All planar embeddings of connected graph G have the same number of faces

A graph with k components, n – e + f = k+1

For simple planar graphs, e(G) ≤ 3n(G) – 6,if G is triangle free, e(G) ≤ 2n(G) – 4pf:2e = ∑L (Fi) ≥ 3f -----(*)f = e – n + 2

=> e ≤ 3n – 6for triangle free case, 3f in (*) -> 4f

K5 and K3,3 are not planar

Maximal Planar Graph

Def. Maximal planar graph: a simple planar graph that is not a spanning subgraph of any other planar graph.

Proposition. The follows are equivalentA)G has 3n-6 edgesB)G is a triangulationC)G is a maximal plane graph

Regular Polyhedra A graph embeds in the plane iff it embeds on a sphere

For a regular polyhedra of degree k and all faces’ length are l e( 2/k + 2/l -1 ) = 2=> (2/k) + (2/l) > 1=> (k – 2)(l – 2) < 4hence k and l can only be k l f

3 3 43 4 64 3 83 5 125 3 20

Graph Theory

Ch6 Planar Graphs(continued)

Kuratowski’s Theorem Theorem. A graph is planar iff it does not contain a subdivisio

n of K5 or K3,3. Kuratowski subgraph: a subgraph contains a subdivision of K

5 or K3,3. minimal nonplanar graph: a nonplanar graph that every prop

er subgraph is planar

Lemma 1if F is the edge set of a face in a planar embedding of G, then G has an embedding with F being the edge set of the unbounded face.

Lemma 2every minimal planar graph is 2-connected.

Lemma 3let S = {x, y} be a separating set of G, if G is nonplanar, there Exist some S-lobe adding (x, y) is nonplanar.

Lemma 4if G is a graph with Fewest Edges among all nonplanar graphs without Kuratowski subgraphs G is 3-connected

Convex embedding: planar embedding that each face boundary is a convex polygon

Theorem. Every 3-connected planar graph has a convex embedding

Theorem. Every 3-connected graph G with at least 5 vertices has an edge e such that G˙e is 3-connected.

Lemma 5. if G has no Kuratowski subgraph, G˙e has no Kuratowski subgraph

Theorem. (Tutte1960) if G is a 3-connected graph without subdivision of K5 or K3,3, then G has a convex embedding in the plane with no three vertices on a line

Pf: induction on n(G)K4:

n(G) > 4:exist e that G˙e is 3-connected. G˙e has no Kuratowski subgraph.e z, H = G˙eH-z is 2-connected.

Definition: H is a minor of G if a copy of H can be obtained by deleting or contracting edges of G.

G is planar iff neither K5 nor K3,3 is a minor of G. Nonseparating Let G be a subdivisions of a 3-connected graph.G is planar iff

every edge e exactly lies in 2 nonseparating cycles.

H-fragment Conflict Planarity testing If a planar embedding of H can be extended to a planar

embedding of G, then in that extension every H-fragment of G appears inside a single face of H.

Planarity testing1. find a cycle G0

2. for each Gi-fragment B, determine all faces of Gi that contain all vertices of attachment of B. call it F(B)3. if F(B) is empty for some B, stop (FAIL). Else, choose one.4. choose a path P between 2 vertices of attachment of B. embed P across F(B). Result in Gi+1.