Maps

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Maps. Graphs on Surfaces. We are mainly interested in embeddings of graphs on surfaces: h : G ! S. An embedding should be differentiated from immersion. On the left we see some forbidden cases for embeddings. Cellular (or 2-cell) embedding.

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Maps

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Maps

Graphs on Surfaces
• We are mainly interested in embeddings of graphs on surfaces:
• h: G ! S.
• An embedding should be differentiated from immersion.
• On the left we see some forbidden cases for embeddings.
Cellular (or 2-cell) embedding
• Embedding h:G ! S is cellular (or 2-cell), if S \ h(G) is a union of open disks.
• A 2-cell embedding is strong (or proper) if the closure of each open disk is a closed disk.
• Proposition: Only connected graphs admit 2-cell embeddings..
• On the left we see two embeddings of K4 in torus S1. The first one is cellular, the second ons is not!
2-Cell Embeddings and Maps
• 2-cell embeddings of graphs are also known as maps. There is a subtile difference in the point of view.
• In the former the emphasis is given to the graph while in the latter the emphasis is in the map, a structure, composed of vertices, edges and faces. Examples of maps include surfaces of polyhedra.
• Maps include different, equivalent, cryptomorphic purely combinatorial definitions that can be used as a foundation of a theory of maps that is independent of topology.
Genus of a Graph
• Let g(G) denote the genus of a graph G. This parameter denotes the minimal integer k, such that G admits an embedding into an orientable surface of genus k.
• Note: (G) = 0 if and only if G is planar.
Euler Characteristics
• To each closed surface S we associate a number (S) called Euler characteristics of S.
• (Sg) = 2 – 2g, for orientable surface of genus g.
• (Nk) = 2 – k, for non-orientable surface of crosscap number (non-orientable genus) k.
Euler Formula
• Let G be a graph with v vertices, e edges cellularly embedded in surface S with f faces. Then
• v – e + f = (S).
Rotation Scheme
• Let G be a connected graph with the vertex set V, with arcs S and edges E. For each v 2 V define the set: S[v] = {s 2 S| i(s) = v}. Let  and  be mappings:
• r: S ! S
• l: S !{-1,+1}.
• with the property:
• Permutation r acts cyclically on S[v], for each v 2 V.
• l(s) = l-1(r(s)), for each s 2 S. [Hence  is a voltage assignment. In our case: l(s) = l(r(s))].
• The triple (G,r,l) is a called a rotation scheme, defining a 2-cell embedding of G into some surface.
Interpretation of Rotation Scheme
• s Ã s0,
• s Ã(s).
• positive Ã True.
• While s  s0 do
• If positive then
• If (s) = 1 then
• s Ã(s)
• else
• positive Ã False;
• s Ã(s)-1
• else
• If (s) = 1 then
• s Ã(s)-1
• else
• positive Ã False;
• s Ã(s)

r(s)

r2(s)

s

r(s)

r3(s)

r4(s)

r(r(s))

r(s)

r2(s)

s

r(s)

r3(s)

r4(s)

r(r(s))

Rotation Scheme and Rotation Projection
• Rotation scheme can be represented by rotation projection.
• Rotation r can be reconstructed from the bottom drawing. Each arc s carries l(s) = 1.
Example
• On the left we see the rotation projection of K4. The faces are triangles.
• There is no cycle with an odd nunber of “crosses”.
• V – E + F = 4 – 6 + 4 = 2.
• The surface is a sphere!
• Exercise: Analyse the faces of the embedding if all crosses are removed from the figure on the left.
Main Fact
• Theorem: Any 2-cell embedding of a graph G into a surface S can be described by a rotation scheme (G,,). Furthermore, by face tracing algorithm the number of faces F can be computed yielding (S). Finally, S is non-orientable if and only if G contains a cycle
• C = (e1,e2, ... , ek) such that
• (C) := (e1) (e2) ... (ek) = -1
Combinatorial Theory of Maps
• There are several cryptomorphic definitions of maps (graphs on surfaces.)
• Rotation schemes represent such a tool.
• Note that we start with a graph G and add additional information (G,,) in order to describe its 2-cell embedding. In some closed surface.
• We may also start directly from maps or polyhedra.

f

v

e

Flag Systems
• Let V,E,F be disjoint (finite) sets.
• Fµ V £ E £ F is a flag system. Here:
• V vertex set,
• E edge set
• F face set.
• A face that is a polygon with d sides, (a d-gon), consists of 2d flags (see figure on the left!)
Flag Systems are General
• Using flag systems we can describe general complexes such as books.
• Note the a 3-book contains a non-orientable Möbious strip.
Flag systems from 2-cell embeddings
• To a 2-cell embedding we associate a flag system as follows. Let V be the set of vertices, E, the set of edges and F the set of faces of the embedding. Define
• µ V £ E £ F as follows:
• (v,e,f) 2 if and only if v, e, and f are pairwise incident.
The 1-skeleton of a flag system.
• Given a flag system µ V £ E £ F, we may study its projection to the first two factors:
• A = {(v,e)| (v,e,f) 2}.
• Define:
• i:A ! V by i: (v,e)  v and
• Ve = {v 2 V| (v,e) 2 A}.
• Assume |Ve| · 2, for each e 2 E.
• We may define r:A ! A by:
• r(v,e) = (w,e) if Ve = {v,w} and
• r(v,e) = (v,e) if Ve = {v}.
• The quadruple (V,A,i,r) is a pre-graph. It is called the 1-skeleton of .
• Given  there is an easy test whether the 1-skeleton is indeed a graph: for each e 2 E we must indeed have |Ve| = 2.
1-co-skeleton
• If we replace the role of V and F in a flag system µ V £ E £ F we obtain a 1-co-skeleton.
• We say that the skeleton and co-skeleton are dual graphs.
Homework

H1: If one of 1-skeleton is a graph is the 1-co-skeleton a graph too? Prove or find a counterexample.

Exercises
• N1. Determine the flag system describing the four-sided pyramid.
• N2. Determine the 1-skeleton and 1-co-skeleton for N1.
• N3. Define the notion of automorphism of a flag system . For the case N1 find the orbits of Aut .
When does a flag system define a surface?
• As we have seen in the case of a book we may have an edge belonging to more than two faces. This clearly violates the rule that each point on a surface has a neighborhood homeomorphic to an open disk.
• Therefore a necessary condition is:
• Each for each flag (v,e,f) 2 there must exist a unique triple (v’,e’,f’) 2 V £ E £ F with v’  v, e’  e, f’  f such that (v’,e,f), (v,e’,f),(v,e,f’) 2.
• Another obvious condition is that the 1-skeleton must be connected.
• However, a flag system satisfying these two conditions may still represent more general spaces than surfaces.
• It may represent a pseudosurface.
• Let us define:
• v = {(f,e)| (v,e,f) 2}.
• e = {(v,f)|(v,e,f) 2}.
• f = {(v,e}| (v,e,f) 2}.
• Each of the three structures defined above can be represented as graph. More presicely, each of them is regular 2-valent graph.
•  is a surface if and only if each graph v, e and f is connected.
Limits of flag systems
• Unfortunately, there are connected graphs whose 2-cell embeddings cannot be represnted by flag systems.
• Proposition. Let G be a connected graph. If G contains a loop or a bridge no 2-cell embedding of G can be described by flag systems.
• [A bridge is an edge whose removal disconnects the graph.]
Some limits of flag systems
• On the left we see K4 embedded in torus with one 4-gon and one 8-con.
• Green and red flag have all three matching components equal.
• This map cannot be described by flag systems.
Self-avoiding maps
• Theorem: A 2-cell embedding of G in some surface can be described by a flag system if an only if neither G nor its dual contains a loop.
• A map that satisifies the conditions of this theorem will be called self-avoiding.
Flags, from a different view-point.
• Let us forget about V,E, F for a moment. Let the set of flags F be given.
• For instance, on the left, we see them as triangles.
• Define the flag graph G(F):
• V() = .
• f ~ f’ if and only if triangles have a common side.
From flags to flag graph.

First the vertices.

From flags to flag graph.

First the vertices.

Next: three kinds of new edges:

along the edges

across the edges.

across the angles.

Flag graphs for 2-cell embedded graphs.

Flag graph  is:

- connected

- trivalent

- contains a 2-factor of form m C4.

Flag graphs for 2-cell embedded graphs.

A practical guide to the construction.

The first step when rectangles are placed on each edge is shown.

Yet another view to flag graphs.
• t0, t1, t2 : F!F
• 02 = 12 = 22 = 1, each fixed-poit free.
• t0t2 = t2t0, also fixed-point free.
• Each invoultion corresponds to a 1-factor. Together they define a cubic graph: the flag graph ().
• The group <0,1,2>, called monodromy group must act transitively on . [This is eaquivalent to saying that () is connected.]
• These axioms define a (combinatorial) map on a surface.
Combinatorial Map.
• Combinatorial map is defined by three involutions satisfying the axioms from the previous slide.
• Orbits of <2,1> acting on  define V.
• Orbits of <0,2> acting on  define E.
• Orbits of <0,1> acting on  define F.
Orientable Map
• Theorem: A map is orientable if and only if the flag graph is bipartite.
Unique Embedding
• Theorem (Whitney): Each 3-connected planar graph admits a unique embedding in the sphere.
• Theorem (Mani). Let Aut G be the group of automorphism of a 3-connectede planar graph G and let Aut M be the group of automorphisms of the corresponding map. Then Aut G = Aut M.
Example - Exercises
• On the left there is an embedding of Q3 on torus.
• N1: Determine the rotation scheme for this embedding.
• N2: Determine the flag graph for this embedding.
Example - Exercises
• On the left there is a different embedding of Q3 on torus.
• N1: Determine the rotation scheme for this embedding.
• N2: Determine the flag graph for this embedding.
• .
Levi graph of a map
• Levi graph of a map M has the vertex set:
• VM t EM t FM,
• Edges are determined by the sides of flags (as triangles).
• WARNING: The graph on the left is not simple!!
Characterisation
• Theorem: Levi graph of a map is simple if neither 1-skeleton nor 1-co-skeleton has a loop.
• Definition: A map M is simple,if and only if its Levi graph is simple.
Homework
• H1: Given Flag graph of a map M, determine whether M is simple! (Prove previous theorem)