Maps

1
Maps
2
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.

3
Cellular (or 2-cell) embedding

• Embedding hG ! 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!

4
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.

5
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.

6
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.

7
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).

8
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 Sv 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 Sv, 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.

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Interpretation of Rotation Scheme

• We follow arcs starting at s0 until we return to
  the initial arc.
• s à s0,
• s à ?(s).
• positive à True.
• While s ? s0 do
• If positive then
• If ?(s) 1 then
• s à ?(s)
• else
• positive à False
• s à ?(s)-1
• else
• If ?(s) 1 then
• s à ?(s)-1
• else
• positive à False
• s à ?(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))
10
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.

11
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.

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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

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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.

14
Flag Systems

• Let V,E,F be disjoint (finite) sets.
• F µ 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!)

15
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.

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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.

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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
• iA ! 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 rA ! 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.

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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.

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Homework

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

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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
  ?.

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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.

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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.

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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.

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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.

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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.

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From flags to flag graph.

• First the vertices.

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From flags to flag graph.

• First the vertices.
• Next three kinds of new edges
• along the edges
• across the edges.
• across the angles.

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Flag graphs for 2-cell embedded graphs.

• Flag graph ? is
• - connected
• - trivalent
• - contains a 2-factor of form m C4.

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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.

30
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.
• t0 t2 t2 t0, also fixed-point free.
• Each invoultion corresponds to a 1-factor.
  Together they define a cubic graph the flag
  graph ?(?).
• The group lt?0,?1,?2gt, called monodromy group
  must act transitively on ?. This is eaquivalent
  to saying that ?(?) is connected.
• These axioms define a (combinatorial) map on a
  surface.

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Combinatorial Map.

• Combinatorial map is defined by three involutions
  satisfying the axioms from the previous slide.
• Orbits of lt?2,?1gt acting on ? define V.
• Orbits of lt?0,?2gt acting on ? define E.
• Orbits of lt?0,?1gt acting on ? define F.

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Orientable Map

• Theorem A map is orientable if and only if the
  flag graph is bipartite.

33
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.

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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.

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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.
• .

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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!!

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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.

38
Homework

• H1 Given Flag graph of a map M, determine
  whether M is simple! (Prove previous theorem)