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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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graphs on surfaces
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
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 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
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
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
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
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
Interpretation of Rotation Scheme
  • We follow arcs starting at s0 until we return to the initial arc.
  • s à s0,
  • s Ã(s).
  • positive à True.
  • While s  s0 do
    • If positive then
      • If (s) = 1 then
        • s Ã(s)
      • else
        • positive à False;
        • s Ã(s)-1
    • else
      • If (s) = 1 then
        • s Ã(s)-1
    • else
        • positive à 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 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
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
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
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.
flag systems

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
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
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
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
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
Homework

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

exercises
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
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
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
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
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
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
From flags to flag graph.

First the vertices.

from flags to flag graph1
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 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 graphs1
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
Yet another view to flag graphs.
  • We may start with three involutions:
      • 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.
  • 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
Orientable Map
  • Theorem: A map is orientable if and only if the flag graph is bipartite.
unique embedding
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
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 exercises1
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
  • 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
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.
homework1
Homework
  • H1: Given Flag graph of a map M, determine whether M is simple! (Prove previous theorem)