Basics on Graph Theory

Graph theory Basics properties Classic problems Fundamental Knowledge
Artiﬁcial Intelligence
Graph theory
G. Guérard
Department of Nouvelles Energies
Ecole Supérieure d’Ingénieurs Léonard de Vinci
Lecture 1
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1 Graph theory
Undirected and directed graphs
Degree
2 Basics properties
Path and Cycles
Speciﬁcations
3 Classic problems
Eulerian circuit
Hamiltonian circuit
Spanning tree
Graph Coloring
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IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction
A graph is a mathematical object consisting of a set of:
NODES - V
EDGES - E
GRAPH is denoted by G = (V , E)
n = |V | and m = |E|
A graph captures pairwise relationship between objects.
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Undirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graphUndirected graph
Undirected graph
An UNDIRECTED GRAPH is a pair: G = (V , E) where V is a
nonempty SET OF VERTICES and E = {(u, v) : u, v ∈ V } is a
SET OF EDGES.
Remark
Let m = (p, q) ∈ E. This means that the edge m connects the
vertex p and the vertex q. Moreover, in such a case p and q are
called the ENDPOINTS of m and we say that p and q are incident
with m, they are ADJACENT or neighbours of each other.
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ExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExampleExample
Undirected graph
Consider the undirected graph G = (V , E) where
V = {a, b, c, d, e}, and
E = {(a, b), (a, c), (a, e), (e, b), (d, b), (c, d), (d, e)}.
Remark
We usually use a graphical representation of the graph. The
vertices are represented by points and the edges by lines
connecting the points.
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Bipartite
Consider a set of discipline D = {A, B, C, D} and students
S = {1, 2, 3, 4}. Every student is interested in some schools
Di ⊂ D. The situation can be easily modelled by the graph
G = (D ∪ S, E).
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Special edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edges
Multiedges
MULTIEDGES are edges which connect the same pair of vertices
and a LOOP connects the vertex with itself. A graph with
multiedges and loops is called a MULTIGRAPH.
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Special edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edgesSpecial edges
We can associated WEIGHTS to edges (see previous example).
These weights can represent cast, proﬁt, length, capacity, action,
etc. of a given connection.
Weight
The weight is a mapping from the set of edges to a set of numbers
or alphabet w : E → R or Σ. The graph with weight function of
numbers is called a NETWORK and denoted by G = (V , E, w).
Alphabet or probability give two others types of graphs: automata and
markov chain.
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Weight
Let us consider the graph G = (V , E, w) seen previously, and
the weight function:
e ∈ E (a, b) (a, c) (a, e) (d, b) (e, b) (d, e) (d, c)
w(e) -9 3 6 0 12 8 0.7
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Directed graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graphDirected graph
Directed graph
A DIRECTED GRAPH is a pair G = (V , E) where V is a
nonempty set of vertices and E = {(u, v) : u, v ∈ V } is a set of
directed edges (or ARCS). If (p, q) ∈ E, p is the head of the arc
and q is the tail of the arc.
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Directed graph
Consider the directed graph G = (V , E) where
V = {a, b, c, d, e}, and
E = {(a, b), (a, c), (a, e), (e, b), (d, b), (c, d), (d, e)}.
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DegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegreeDegree
Undirected graph
The DEGREE of the vertex v is the number of edges incident to the
vertex, with loops counted twice. The degree of a vertex is
denoted deg(v) or Γ(v). A vertex v with degree 0 is called an
ISOLATED. A vertex with degree 1 is called an endvertex or a
LEAF.
The degree of the graph G, (G) is the maximum degree of its
vertices.
(G) = max
v∈V
Γ(V ).
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Degree
Consider the graph
Isolated vertices are x5 and x7. Leaves are x4 and x6.
(G) = max
v∈V
Γ(V ) = deg(x2) = 5.
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Directed graph
The IN-DEGREE of a vertex v, Γ−(v), is the number of arc
incoming. The OUT-DEGREE of a vertex v, Γ+(v), is the number
of arc going out. The degree of a vertex is
Γ(v) = Γ−(v) + Γ+(v).
A vertex s for which Γ−(v) = 0 and Γ+(v) > 0 is called a
SOURCE and a vertex t for which Γ−(v) > 0 and Γ+(v) = 0 is
called a SINK.
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Degree
Consider the directed graph
Γ−(x) = 2, Γ+(x) = 3; Γ−(y) = 5, Γ+(y) = 1.
(G) = max
v∈V
Γ(V ) = deg(y) = 6. There is no source or sink
(technically, y can be called a sink because we can’t escape from
y to another vertex).
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Source and sink
Consider the directed graph
Sources are vertices a and c, sink is the vertex e.
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Theorem
For any graph, we have ∑
v∈V
deg(v) = 2|E|. The demonstration is
trivial, any edge connects two vertices. So an edge (directed or
undirected) increase the total of degrees by two.
Theorem
The number of vertices of odd degree is even. The sum of all the
degrees is equal to twice the number of edges. Since the sum of the
degrees is even and the sum of the degrees of vertices with even degree
is even, the sum of the degrees of vertices with odd degree must be
even. If the sum of the degrees of vertices with odd degree is even,
there must be an even number of those vertices.
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DefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinitionDefinition
Path
A PATH (or CHAIN in a directed graph) in a graph is a sequence
of edges (or arcs) which connect a sequence of vertices. To be more
specific, a sequence (e1, . . . , en) is called a path of the LENGTH n
if there are vertices v0, . . . , vn such that ei = (vi−1, vi ), i = 1..n.
The first vertex v1 is called START vertex, and the last vertex vn
is called END vertex. Both of them are called TERMINAL vertices
of the path. If v0 = vn then the sequence is called a closed path.
A closed path which the edges and vertices are distinct (exept
terminals) is called a CYCLE (or a CIRCUIT in directed graph).
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Path
Consider the graph
(d, e, g, b, a) is a closed path. (d, g, b, a) is a cycle.
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Acyclic
A graph without cycles is called ACYCLIC.
Connected
A graph is called CONNECTED if for any couple of vertices u
and v there exists a path connecting u and v, i.e. with start vertex
is u and end vertex is v.
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TreeTreeTreeTreeTreeTreeTreeTreeTreeTreeTreeTreeTreeTreeTreeTreeTree
Tree
A connected and acyclic graph is called a TREE.
a, b, c, d are trees, e has a cycle.
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Complete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graphComplete graph
Complete graph
A COMPLETE graph Kn is the graph with n vertices and all the
pairs of vertices are adjacent to each other.
Theorem
In any Kn, deg(v) = n − 1, ∀v ∈ V . The number of edges is
|E| =
[
i= 1]n∑i = n(n−1)
2 .
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SubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraphSubgraph
Subgraph
A graph H = (VH, EH ) is called a SUBGRAPH of G = (V , E)
when VH ⊂ V and EH ⊂ E.
Consider graphs
G1 and G2 are subgraphs of G.
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Eulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuitEulerian circuit
Problem
GIVEN THE GRAPH, IS IT POSSIBLE TO CONSTRUCT A PATH (OR
A CYCLE) WHICH VISITS EACH EDGE EXACTLY ONCE?
Eulerian circuit
How can we recognize Eulerian graph? They have two
characterizations: node degrees, existence of a special collection
of cycles.
Theorem
For a connected graph G, the following statements are
equivalent: G is Eulerian (1); Every vertex of G has even degree
(2); The edges of G can be partitioned into edge-disjoint cycles (3).
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Proof 1 > 2
Assume G is Eulerian ⇐⇒ there exists a circuit that includes
every edge of G. Every time a circuit enters a node v on an
edge, it leave on a different edge. Since the circuit never repeats
an edge, the number of edges incident with v is even ⇒ deg(v)
is even.
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Proof 2>3
Suppose every node of G has even degree. We use induction on
the number of cycles in G. G is connected and without nodes of
degree 1, so G is not a tree, Ghas at least one cycle Cn1 . Let G
be the graph obtained by removing Cn1 from G. All edges of G
have even degree and we can proceed recursively to prove that
G can be partitioned into cycles Cn2 . . . Cnk
. Then, Cn1 . . . Cnk
is
a partition of G into edge-disjoint cycles.
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Proof 3>1
Suppose that the edges of G can be partitioned into k edge-disjoint cycles
Cn1 . . . Cnk
. Because G is connected, every such cycle is an Eulerian circuit
which must share a node with another cycle ⇒ these circuits can be patched
until we obtain one Eulerian circuit which is the whole graph G.
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Eulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian pathEulerian path
Question
How to recognize graph which contain an Eulerian path?
Note that: If the graph is Eulerian, then it contains an Eulerian path
because every Eulerian circuit is also a path.
Corollary
A connected graph G contains an Eulerian path iff there are at
most two vertices of odd degree.
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Hamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuit
Problem
GIVEN THE GRAPH, IS IT POSSIBLE TO CONSTRUCT A PATH (OR
A CYCLE) WHICH VISITS EACH VERTEX EXACTLY ONCE?
Hamiltonian circuit
How can we recognize Eulerian graph? There is no simple
characterisation for this problem. Hamiltonian graphs can have
all even degrees, all odd degrees, or a mixture.
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Hamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuitHamiltonian circuit
Dirac’s Theorem
Let G be a graph of order n ≥ 3. If deg(G) ≥ n
2 , then G is
Hamiltonian.
Dirac’s Theorem 2
Let G be a graph of order n ≥ 3. If deg(x) + deg(y) ≥ n for all
pairs of nonadjacent nodes x, y, then G is Hamiltonian.
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Spanning tree
A SPANNING TREE T of an undirected graph G is a subgraph that is a
tree which includes all of the vertices of G. A MINIMUM SPANNING TREE
(MST) connects all the vertices together with the minimal total weighting
for its edges (see KRUSKAL and PRIM).
Applications
In order to minimize the cost of power networks, wiring connections, piping,
automatic speech recognition, etc., we use algorithms that gradually build a
spanning tree (or many such trees) as intermediate steps in the process of
ﬁnding the minimum spanning tree.
The Internet and many other telecommunications networks have
transmission links that connect nodes together in a mesh topology that
includes some loops. In order to avoid bridge loops and routing loops, many
routing protocols designed for such networks require each router to
remember a spanning tree.
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Kruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s Algorithm
KRUSKAL’S ALGORITHM is a greedy algorithm for the MST
problem.
Initially, let T ← ∅ be the empty graph on V .
Examine the edges in E in increasing order of weight:
If an edge connects two unconnected components of T, then add
the edge to T
Else, discard the edge and continue.
Terminate when there is only one connected component.
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Prim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s Algorithm
PRIM’S ALGORITHM is a greedy algorithm for the MST
problem.
Initialize a tree with a single vertex, chosen arbitrarily from the graph.
Grow the tree by one edge: of the edges that connect the tree to vertices
not yet in the tree, ﬁnd the minimum-weight edge, and transfer it to
the tree.
Repeat step 2 (until all vertices are in the tree).
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Deﬁnition
GRAPH COLORING is a special case of graph labeling; it is an
assignment of labels traditionally called "colors" to elements of a
graph subject to certain constraints. In its simplest form, it is a
way of coloring the vertices of a graph such that no two adjacent
vertices share the same color; this is called a VERTEX COLORING.
Similarly, an EDGE COLORING assigns a color to each edge so
that no two adjacent edges share the same color, and a face coloring
of a planar graph assigns a color to each face or region so that
no two faces that share a boundary have the same color.
Without any qualiﬁcation, a coloring is always a vertex coloring.
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Deﬁnition
Given a graph G(V , E) with vertex set V and edge set E, a k-colouring of G
is a function C : V → {1 . . . k} that assigns distinct values to adjacent
vertices. For each edge e = (u, v) ∈ E, we require C(u) = C(v). If G has a
k-colouring then it is said to be k-colourable. In words, a graph is
k-colourable if one can draw it in such a way that no two adjacent vertices
have the same colour. The chromatic number χ(G) is the smallest number k
such that G is k-colourable.
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ApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplicationsApplications
The problem of coloring a graph arises in many practical areas
such as pattern matching, sports scheduling, designing seating
plans, exam timetabling, the scheduling of taxis, and solving Sudoku
puzzles.
Example
A given set of jobs need to be assigned to time slots, each job requires
one such slot. Jobs can be scheduled in any order, but pairs of jobs
may be in conflict in the sense that they may not be assigned to the
same time slot, for example because they both rely on a shared
resource. The corresponding graph contains a vertex for every job
and an edge for every conflicting pair of jobs. The chromatic number
of the graph is exactly the minimum makespan, the optimal time to
finish all jobs without conflicts.
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Fundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledgeFundamental knowledge
YOU HAVE TO KNOW BEFORE THE TUTORIAL:
1 Caracteristics and differences between undirected and directed
graphs;
2 Notion of degrees, how to calculate them;
3 Notion of path and cycles;
4 Hamiltonian and eulerian deﬁnitions (Problem block);
5 Kruskal and Prim’s algorithms;
6 Graph coloration.
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Basics on Graph Theory

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