Introduction to graph theory and applications

1
Introduction to graph theory and applications

• Introduction and definitions
• Typical network modelling applications
• Classic graph theory problems and proofs
• The Seven Bridges of Königsburg
• The four colour map colouring theorem
• The three cottage problem
• Data structures used for storing graphs
• Incidence and adjancancy lists and matrixes

2
Definitions

• Graph theory concerns the study of networks based
  on a mathematical abstraction of the form of a
  graph. A graph is made up of vertices (singular
  vertex) and edges. An edge connects exactly 2
  vertices. These are the same vertex in the
  special case where the edge is called a loop or
  loopback.
• A vertex can have any number of edges connected
  to it.
• Edges might be directed, and drawn as an arrow to
  indicate that network flow or traffic or the
  connection represented by the edge works in one
  direction only. In an undirected graph the edges
  are bidirectional. Edges may be associated with a
  numeric cost. The meaning of edge cost will
  depend upon the graph application.

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Vertices and bidirectional edges
4
More definitions

• A path is a route through a graph visiting
  vertices and edges in turn. A cycle is a path
  that ends at the starting vertex. A Hamiltonian
  path or cycle visits all vertices exactly once. A
  Eulerian path or cycle visits all edges exactly
  once.
• A graph is simple if no edge is a loop, and no 2
  edges have the same endpoints.
• A planar graph can be drawn on a plane surface,
  e.g. a piece of paper, so that no edges cross
  over other edges.

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Even more definitions ...

• A directed acyclic graph is a directed graph
  without cycles, i.e. you can't get back to the
  vertex where you started by following edges in
  their defined direction. This is a way to map a
  forest of trees so that subtrees can be shared
  between trees. Sources are vertices all of whose
  edges lead out from them and sinks are vertices
  all of whose edges lead into them. A tree is a
  graph which is connected, acyclic and simple.

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Terms to remember

• graph
• vertex (pl vertices)?
• edge
• edge cost
• undirected graph
• directed graph
• simple graph
• loop

• path
• cycle
• Hamiltonian
• Eulerian
• planar graph
• acyclic directed graph
• forest
• tree

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Typical Graph Applications 1

• Modelling a road network with vertexes as towns
  and edge costs as distances.
• Modelling a water supply network. A cost might
  relate to current or a function of capacity and
  length. As water flows in only 1 direction, from
  higher to lower pressure connections or downhill,
  such a network is inherently an acyclic directed
  graph.
• Modelling the recent contacts of someone who has
  become ill with a notifiable illness, e.g. Sars
  or Meningitis. Edge costs might be a function of
  the probability that the contact resulted in an
  infection.

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Typical Graph Applications 2

• Dynamically modelling the status of a set of
  routes by which traffic might be directed over
  the Internet.
• Modelling the connections between a number of
  potential witnesses or suspects who were reported
  or came forward as having been within the
  vicinity of a serious crime within an hour of of
  when it occurred.
• Minimising the cost and time taken for air travel
  when direct flights don't exist between starting
  and ending airports.
• Using a directed graph to map the links between
  pages within a website and to analyse ease of
  navigation between different parts of the site.

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Classic Graph Theory Problems

• Graph theory started from a mathematical
  curiosity.
• "The Seven Bridges of Königsberg is a problem
  inspired by an actual place and situation. The
  city of Kaliningrad, Russia (at the time,
  Königsberg, Germany) is set on the Pregolya
  River, and included two large islands which were
  connected to each other and the mainland by seven
  bridges. The question is whether it is possible
  to walk with a route that crosses each bridge
  exactly once, and return to the starting point.
  In 1736, Leonhard Euler proved that it was not
  possible."
• Source http//en.wikipedia.org/wiki/Seven_Bridges
  _of_Koenigsberg

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Seven Bridges of Königsberg 2

• "In proving the result, Euler formulated the
  problem in terms of graph theory, by abstracting
  the case of Königsberg -- first, by eliminating
  all features except the landmasses and the
  bridges connecting them second, by replacing
  each landmass with a dot, called a vertex or
  node, and each bridge with a line, called an edge
  or link. The resulting mathematical structure is
  called a graph."

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Seven Bridges of Königsberg 3

• "The shape of a graph may be distorted in any way
  without changing the graph itself, so long as the
  links between nodes are unchanged. It does not
  matter whether the links are straight or curved,
  or whether one node is to the left of another.
• Euler realized that the problem could be solved
  in terms of the degrees of the nodes. The degree
  of a node is the number of edges touching it in
  the Königsberg bridge graph, three nodes have
  degree 3 and one has degree 5. Euler proved that
  a circuit of the desired form is possible if and
  only if there are no nodes of odd degree. Such a
  walk is called an Eulerian circuit or an Euler
  tour. Since the graph corresponding to Königsberg
  has four nodes of odd degree, it cannot have an
  Eulerian circuit."

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Seven Bridges of Königsberg 4

• "The problem can be modified to ask for a path
  that traverses all bridges but does not have the
  same starting and ending point. Such a walk is
  called an Eulerian trail or Euler walk. Such a
  path exists if and only if the graph has exactly
  two nodes of odd degree, those nodes being the
  starting and ending points. (So this too was
  impossible for the seven bridges of Königsberg.)"

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The four colour theorem

• This theorem states that given any set of areas
  on a planar surface, e.g. representing political
  regions on a map, it is possible to colour every
  area so that no 2 regions sharing a border need
  use the same colour if 4 colours are used.
• Sharing a border means a linear border between 2
  areas of more than zero length e.g. many regions
  can meet at a point but this doesn't count as a
  border. Also an area within the set must be
  single and contiguous.

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The four colour theorem 2

• Political geographic entities such as countries
  sometimes break this requirement by having
  enclaves, e.g. the USA includes Alaska which is
  not part of the largest geographically contiguous
  region of the USA. Old maps of English and Welsh
  counties split the county of Flint into 2
  seperate areas.
• This theorem comes from a question (conjecture)
  asked by a student Francis Guthrie of his maths
  lecturer Augustus De Morgan in 1852. It was
  finally proved in 1976.

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Graph theory presentation of the theorem

• "To formally state the theorem, it is easiest to
  rephrase it in graph theory. It then states that
  the vertices of every planar graph can be colored
  with at most four colors so that no two adjacent
  vertices receive the same color. Or "every planar
  graph is four-colorable" for short. Here, every
  region of the map is replaced by a vertex of the
  graph, and two vertices are connected by an edge
  if and only if the two regions share a border
  segment (not just a corner)"
• Source http//en.wikipedia.org/wiki/Four-color_th
  eorem

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Three cottage problemSource http//en.wikipedia
.org/wiki/Three_cottage_problem

• "The three cottage problem is a well-known
  mathematical puzzle. It can be stated like this
• Suppose there are three cottages on a plane (or
  sphere) and each needs to be connected to the
  gas, water, and electric companies. Is there a
  way to do so without any of the lines crossing
  each other?"

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Three cottage problem 2

• "The problem is part of the mathematical field of
  topological graph theory which studies the
  embedding of graphs on surfaces. In more formal
  graph theoretic terms the problem asks whether
  the complete bipartite graph K3,3 is planar.
  Kazimierz Kuratowski proved in 1930 that K3,3 is
  nonplanar, and thus that the three cottage
  problem has no solution.
• But K3,3 is toroidal, that is it can be embedded
  on the torus. In terms of the three cottage
  problem this means the problem can be solved by
  punching a hole through the plane (or the
  sphere). This changes the topological properties
  of the surface and using the hole we can connect
  the three cottages without crossing lines."

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Three cottage problem 3

• From this description note that the term
  "complete bipartite graph" means a graph with 2
  sets of vertices where every vertex in one set is
  connected by an edge to every vertex in the other
  set. The designation K3,3 for this graph
  indicates that there are 3 vertices in each of
  the 2 sets, representing utilities and cottages
  respectively. K5 is diagrammed on the right.

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Three cottage problem 4

• Kuratowski's proof is based on a process
  involving simplification of complex graphs into
  simpler graphs by removing vertices and edges in
  certain situations and finding instances of one
  of the 2 simplest non planar graphs as subsets of
  the more complex graph. The other simplest
  non-planar graph is a complete graph known as K5,
  because it has a single set of 5 vertices with
  each vertex connected to every other vertices.

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Data structures used for storing graphs

• Before choosing one of various options or
  designing a customised approach, programmers need
  to consider application data, and the means by
  which this will be attached to vertices and
  edges. Records will need to be attached either to
  vertices, or to edges, or one type of record
  might need attachment to vertices and another to
  edges. The choice and design of data structure(s)
  will influence and be influenced by program
  performance and memory usage requirements.

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Incidence list

• The connectivity of an edge is regular in that
  each represents a connection between exactly 2
  vertices. This lends itself to fixed-size record
  designs, e.g. an array of records known as an
  incidence list.
• The coding in the next slide uses pointers to the
  vertex records, as a vertex may appear in any
  number of edge records. Alternatively, if the
  vertices are stored in an array which is
  populated before the edge records are created,
  and unchanged during the runtime of the
  application, the edge records could store the
  integer array indices. Which approach is optimal
  depends upon how vertex records are stored and
  how this collection of data is searched and
  updated.

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Incidence list structure coding
23
Adjacency list

• The connectivity of a vertex is irregular, in the
  sense each can be connected using a variable
  number of edges. Vertex information can be stored
  in a fixed width record containing the head
  pointer of a linked list. Each linked list item
  will represent an edge. The latter will contains
  references (e.g. array indices, keys or pointers)
  to edge record storage so this isn't duplicated.
  If this isn't done within the edge records
  themselves, the linked list nodes must also
  reference other vertices connected via the edges.
  This approach is known as an adjacency list.

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Adjacency list structure coding
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Adjacency list coding comments

• The above structure is better suited to a
  directed graph than an undirected graph. A link
  from a webpage to another page (i.e. a directed
  edge as in the above example ) does not require a
  link in the opposite direction. Using this kind
  of approach for an undirected graph would either
  lead to duplication of information, in the sense
  that each edge would have to be stored and
  updated twice, or added complexity of path access
  and traversal. Edge costs, if required, could be
  stored as an additional field within the url_list
  structure, for which each node in the list is an
  edge. A reason for having edge costs in this
  example might be if not all links between pages
  are equally easy for a user to find.

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Incidence matrix

• This structure is a 2 dimensional array where the
  rows index the edges, and columns index the
  vertices. In the simplest implementation the
  array element is boolean, with a 1 indicating a
  connection and a 0 indicating no connection. The
  advantage of this structure is that it enables
  data to be rapidly indexed, either for vertices
  or for edges. The disadvantage is that memory
  proportional to V x E is required, where V is the
  number of vertices, and E the number of edges.
  List or array based structures require memory
  proportional to V E.

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Adjacency matrix

• This is a V x V 2D array where V is the number of
  vertices. In the simplest case, if the data
  present at index ij is a 1 this indicates a
  (possibly directed) edge from i to j, while a 0
  indicates there being no such edge. If the edges
  have costs, this could be a floating point number
  present at index ij, with a sentinel value,
  e.g. 0 or -1, indicating lack of an edge from
  vertex i to vertex j. This data structure makes
  it easier to search for subgraphs of particular
  patterns or identities, e.g. K3,3 or K5.

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Conclusions

• Graph theory enables us to study and model
  networks and solve some difficult problems
  inherently capable of being modelled using
  networks.
• Various terms e.g. vertex and edge, are
  associated with graph theory which gives these
  terms special meanings. These meanings need to be
  understood and remembered in order to apply graph
  theoretic approaches to solving problems.
• When solving a problem by developing a
  graph-based program, careful attention must be
  given at the design stage to the structuring of
  data to help make solving the problem tractable,
  to enable linkages to be traced efficiently and
  to avoid duplication of data.