DEA in Bioinformatics 2002-2003
Principles of graph theory
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Proverbios y cantares - XXIX
Caminante, son tus huellas
el camino y nada más;
Caminante, no hay camino,
se hace camino al andar.
Al andar se hace el camino,
y al volver la vista atrás
se ve la senda que nunca
se ha de volver a pisar.
Caminante no hay camino
sino estelas en la mar.
Antonio Machado
Contents

Basic concepts of graph theory







Applications of graph theory to metabolic networks







Definitions
Descriptions of a graph
Walks, trails and paths
Trees
Spanning trees
Structural properties of a graph
Examples of metabolic networks
Mapping metabolic networks onto a graph
Taversal rules for metabolic graphs
Path finding
Pathway reconstruction by reaction clustering
From gene expression data to pathways
Acknowledgements
DEA in Bioinformatics 2001
Basic concepts
of graph theory
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Basic concepts of graph theory
Graph definitions
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Graph
graph
vertex (node)
edge
self-loop
proper edge
multi-edge
simple graph


A graph (G) contains a set of
vertices (V) and a set of edges
(E)
A simple graph contains no selfloop and no multi-edge
Directed Graph (= Digraph)
digraph
vertex (= node)
arc (= directed edge)
self-loop
proper arc
multi-arc
not a multi-arc !

simple digraph


A directed edge (or arc) is
characterized by a head and a
tail
A digraph is a graph whose
edges are directed
A partially directed graph is a
graph combining directed and
non-directed edges
Basic concepts of graph theory
Graph descriptions
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Graph descriptions : incidence matrix




one row per edge
one colum per vertex
value = 1 if edge and vertex are incident
Problems


only valid for undirected graphs
inefficient storage (many empty cells)
edge\vertex
e1
e2
e3
e4
e5
e6
e7
e8
a
1
1
0
0
0
0
0
0
b
1
1
1
1
1
0
0
0
c
0
0
1
0
0
1
0
0
d
0
0
0
0
0
0
0
0
e
0
0
0
1
1
1
1
0
f
0
0
0
0
0
0
0
0
g
0
0
0
0
0
0
0
1
h
0
0
0
0
0
0
0
1
e1
a
e2
e5
e7
d
f
b
e3
e4
c
e6
e
g
e8
h
Graph descriptions : adjacency matrix





one row per vertex
one colum per vertex
value = 1 if vertices are adjacent
diagonal = self-loops
problems


from\to
a
b
c
d
e
f
g
h
no possibility to represent multi-arcs
inefficient storage (many empty cells)
a
0
0
0
0
0
0
0
0
b
1
0
0
0
1
0
0
0
c
0
1
0
0
1
0
0
0
d
0
0
0
1
0
0
0
0
e
0
1
0
0
0
0
0
0
f
0
0
0
0
0
0
0
0
g
0
0
0
0
0
0
0
0
h
0
0
0
0
0
0
1
0
e1
a
e2
e3
e5
e7
d
f
b
e4
c
e6
e
g
e8
h
Graph descriptions : adjacency list



A list of out-going vertices is associated to
each vertex
Compact representation
Optionally, a list of in-going vertices can be
added to allow reverse-traversal of the
graph
Vertex
a
b
c
d
e
f
g
h
out
(b,b)
(c,e)
()
(d)
(b,c)
()
(h)
()
in
()
(a,e)
(b,e)
(d)
(b)
()
()
(h)
e1
a
e2
e3
e5
e7
d
f
b
e4
c
e6
e
g
e8
h
Graph descriptions : formal description




one row per edge
one column for heads
one column for tails
optional columns for edge
attributes
(label, weight, color, …)
e1
a
e2
e3
e5
e7
d
f
b
e4
c
e6
e
g
e8
h
head tail label
a
b
e1
a
b
e2
b
c
e3
b
e
e4
e
b
e5
e
c
e6
d
d
e7
g
h
e8
Basic concepts of graph theory
Walks, trails and paths
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Walk


A walk from vertex A to vertex B is an alternating sequence of
vertices and edges, representing a continuous traversal from A
to B
Remarks



a
r
A walk can be described unequivocally by the sequence of edges
(e.g.: d, e, a, d, n,p,h,t,t,t)
In a non-simple graph (i.e. with multi-edges), a walk is not
described unequivocally by a sequence of vertices
An edge or a vertex can appear repeatedly in the same walk
(e.g.: edges d and t , and vertices a, e, c on the figure)
a
b
b
g
d e
c
f
e
d
f i g j h
m n
j o
i
k l
p
l
q
k
s
t
c
h
m
Closed walk
A closed walk is a walk whose initial and final vertices
are identical
(e.g.: d, e, a, d, n,p,h,t,t,t,b,a)

a
r
a
b
b
g
d e
c
f
e
d
f i g j h
m n
j o
i
k l
p
l
q
k
s
t
c
h
m
Trail


a
r
A trail is a walk with no repeated edges
(e.g.: d, e, a, c,l,q,h,t)
Remark: a vertex can appear repeatedly in the same
trail (e.g.: a and c on the figure)
a
b
b
g
d e
c
f
e
d
f i g j h
m n
j o
i
k l
p
l
q
k
s
t
c
h
m
Path

a
r
A path is a trail with no repeated vertices, except
possibly the initial and final vertex
(e.g. c,l,q,h)
a
b
b
g
d e
c
f
e
d
f i g j h
m n
j o
i
k l
p
l
q
k
s
t
c
h
m
Path

a
r
A cycle is a closed path with at least one edge
(e.g. c,l,q,h,b,a)
a
b
b
g
d e
c
f
e
d
f i g j h
m n
j o
i
k l
p
l
q
k
s
t
c
h
m
Connected graph

a connected graph is a graph in which there is a walk between
every pair of distinct vertices
Connected graph
a
r
a
b
b
g
d e
c
f
e
d
f i g j h
m n
j o
i
k l
p
l
q
k
s
Non-connected graph
t
a
c
c
d
h
r
k
a
b
b
g
d e
f
e
f i g j h
m n
j o
i
l
m
t
c
q
k
m
s
Basic concepts of graph theory
Trees
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Tree

a tree is a connected graph that has no cycles
Tree
a
b
a
d
e
c
d
k
m
li
l
k
f
n
b
f
i g
c
j h
j
p
m
Rooted tree
A rooted tree is a directed tree having a distinguished vertex r
called the root such that for each other vertex v, there is a
directed path from the root to v
Each non-root node has a single parent


Rooted tree
root=a
d
c
d
k
e
m
li
l
k
d
b
e
f
f
j
Depth
root = a
c
i g
j h
o
e
m
b
h
d
k
e
i
0
c
k
1
l
l
2
f
f
p
3
i
m
o
j
g
4
j
h
5
p
m
6
c
7
h
Queue and stack

A queue is a sequence of elements
such that each new element is
added (enqueued) to one end,
called the back of the queue, and an
element is removed (dequeued)
from the other end, called the front

A stack is a sequence of elements
such that each new element is
added (or pushed) onto one end,
called the top, and an element is
removed (popped) from the same
end
Level-order tree traversal with a queue


Enqueue root
While queue is not empty


Dequeue a vertex and write it to the output list
Enqueue its children left-to-right
a
e
d
i
b
k
f
g
j
m
c
h
l
Step
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Output
a
e
d
i
b
k
l
f
g
j
h
m
c
Queue
a
e,d
d,i,b
i,b,k,l
b,k,l
k,l,f
l,f
f
g
j,h
h,m
m,c
c
1
2
3
4
5
6
8
9
10
12
13
11
7
Pre-order tree traversal with a stack


Push root onto the stack
While stack is not empty


Pop a vertex off stack, and write it to the output list
Push its children right-to-left onto stack
a
e
d
i
b
k
f
g
j
m
c
h
l
Step
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Output
a
e
i
b
f
g
j
m
c
h
d
k
l
Stack
a
d,e
d,b,i
d,b
d,f
d,g
d,h,j
d,h,m
d,h,c
d,h
d
l,k
l
1
2
11
3
4
12 13
5
6
7
8
9
10
Basic concepts of graph theory
Spanning trees
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Tree in a graph



A tree T in a graph G is a connected subgraph which contains
no cycle
The edges and vertices of T are called tree-edges and treevertices
A frontier-edge is a non-tree edge with one endpoint in T and
one endpoint not in T.
a
b
a
c
d
d
e
m
k
k
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
Spanning tree




A spanning tree is a tree which contains all the vertices of a graph
A spanning tree does not necessarily contain all the edges
A fundamental cycle in the graph G is the unique cycle which is
created when a non-tree edge is added to the spanning tree T
Each non-tree edge corresponds to a fundamental cycle in the graph
a
b
a
c
d
d
e
m
k
k
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
Depth-first-search (DFS)




Initialize tree at a given vertex (for example a)
Initialize the set of frontier edges as empty
Set dfnumber(a) to 0
Initialize label counter i to 1
b
a
a (0)
c
d
d
e
m
k
k
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=1
Depth-first-search

While T does not span G

update the set of frontier edges
b
a
a (0)
c
d
d
e
m
k
k
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=1
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b
a
a (0)
c
d
d (1)
e
m
k
k
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=2
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b
a
a (0)
c
d
d (1)
e
m
k
k (2)
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=3
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b
a
a (0)
c
d
d (1)
e
f
k (2)
l
i
g
i
n
m
k
f
e
h
g
j
o
j
l (3)
c
b
h
p
q
m
i=4
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b
a
a (0)
c
d
d (1)
e
f
k (2)
l
i
g
i
n
m
k
f
e
h
g
j
o
j
l (3)
c
b
h
p
q
m (4)
i=5
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b
a
a (0)
c
d
d (1)
e
f
k (2)
l
i
g
i
n
m
k
f
e
h
g
j
o
j
l (3)
c (5)
b
h
p
q
m (4)
i=6
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b (6)
a
a (0)
c
d
d (1)
e
f
k (2)
l
i
g
i
n
m
k
f
e
h
g
j
o
j
l (3)
c (5)
b
h
p
q
m (4)
i=7
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b (6)
a
a (0)
c
d
e
e (7)
d (1)
k (2)
l
i
g
i
n
m
k
f
f
h
g
j
o
j
l (3)
c (5)
b
h
p
q
m (4)
i=8
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b (6)
a
a (0)
c
d
e
e (7)
d (1)
k (2)
l
i (8)
l (3)
g
i
n
m
k
f
f
c (5)
b
h
g
j
o
j
h
p
q
m (4)
i=9
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b (6)
a
a (0)
c
d
e
e (7)
d (1)
n
m
k
k (2)
l
f
f
i (8)
l (3)
c (5)
b
g
i
h
g
j
o
j (9)
h
p
q
m (4)
i=10
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b (6)
a
a (0)
c
d
e
e (7)
d (1)
k (2)
l
f
n
m
k
f
i (8)
l (3)
c (5)
b
g (10)
i
h
g
j
o
j (9)
h
p
q
m (4)
i=11
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b (6)
a
a (0)
c
d
e
e (7)
d (1)
k (2)
l
f
n
m
k
f
i (8)
l (3)
c (5)
b
g (10)
i
h (11)
g
j
o
j (9)
h
p
q
m (4)
i=12
Depth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge whose labelled endpoint has the largest possible
dfnumber
add this edge to the tree
select the unlabelled endpoint of this edge, and set its dfnumber to i
i := i+1
b (6)
a
a (0)
c
d
e
e (7)
d (1)
n
m
k
k (2)
l
f
f (12)
i (8)
l (3)
c (5)
b
g (10)
h (11)
g
j
i
o
j (9)
h
p
q
m (4)
i=13
Breadth-first-search (BFS)




Initialize tree at a given vertex (for example a)
Initialize the set of frontier edges as empty
Write label 0 on vertex a
Initialize label counter i to 1
b
a
a (0)
c
d
d
e
m
k
k
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=1
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
b
a
a (0)
c
d
d (1)
e
m
k
k
l
i
l
f
e
f
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=2
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b
a
d
e (2)
f
m
k
k
l
f
e
i
l
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=3
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f
m
k
k
l
f
e
i
l
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=4
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f
m
k
k (4)
l
f
e
i
l
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=5
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f
m
k
k (4)
l
f
e
i
l (5)
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=6
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f
m
k
k (4)
l
f
e
i (6)
l (5)
g
i
n
c
b
h
g
j
o
j
h
p
q
m
i=7
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f
m
k
k (4)
l
f
e
i (6)
l (5)
n
c
b
g
i
o
j (7)
h
g
j
h
p
q
m
i=8
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f (8)
m
k
k (4)
l
f
e
n
i (6)
l (5)
c
b
g
h
g
j
i
o
j (7)
h
p
q
m
i=9
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f (8)
m
k
k (4)
l
f
e
n
i (6)
l (5)
c (9)
b
g
h
g
j
i
o
j (7)
h
p
q
m
i=10
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f (8)
m
k
k (4)
l
f
e
n
i (6)
l (5)
c (9)
b
g
h
g
j
i
o
j (7)
h
p
q
m (10)
i=11
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f (8)
m
k
k (4)
l
f
e
n
i (6)
l (5)
c (9)
b
g (11)
h
g
j
i
o
j (7)
h
p
q
m (10)
i=12
Breadth-first-search

While T does not span G





update the set of frontier edges
select a frontier edge for which the labeled endpoint has the smallest
possible label
add edge to the tree T
select the unlabelled vertex of the edge and set its label to i
increment i (i := 1+1)
a (0)
c
d (1)
b (3)
a
d
e (2)
f (8)
m
k
k (4)
l
f
e
n
i (6)
l (5)
c (9)
b
g (11)
h (12)
g
j
i
o
j (7)
h
p
q
m (10)
i=13
Result comparison : DFS versus BFS
b (6)
c (5)
b
a
a (0)
c
de (7) e
d (1)
m
k
n
f
f (12)
i (8)
l
g (10)
g
j
i
o
j (9)
h
q
m (4)
b (3)
c
d (1)
d
e
e (2)
k
k (4)
l
n
i (6)
l (5)
g (11)
f
f (8)
m
c (9)
b
a
a (0)
DFS
p
l (3)
k (2)
h (11)
h (12)
g
j
i
o
j (7)
h
p
q
m (10)
BFS
Dijkstra's Shortest path finding algorithm


Input: a weighted connected graph G whose edge-weights are nonnegative, and a starting vertex a
Output: a spanning tree, rooted at a, whose path from each vertex v
is the shortest path from a to v in G; the vertex-labelling gives the
distance from s to each vertex
b
7
a
2
3
d
e
k
7
2
f
4
3
6
2
3
i
l
c
8
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm



Initialise the Dijkstra tree T as vertex a
dist[a] = 0
write label 0 on vertex a
b
7
a (0)
2
3
d
e
k
7
2
f
4
3
6
2
3
i
l
c
8
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G
Update frontier edges
For each frontier edge e


•
•
let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
a (0)
3 (3)
e
2 (2)
d
k
7
2
3
i
l
c
8
2
f
4
3
6
b
7 (7)
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G
Update frontier edges
For each frontier edge e


•
•
let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
Let e be a frontier edge that has the smallest P-value
Add edge e to T, and set dist[y] = P(e)


a (0)
3 (3)
e
2 (2)
d (2)
k
7
2
f
i
l
c
8
2
3
4
3
6
b
7 (7)
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G


Update frontier edges
For each frontier edge e
•
•


let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
Let e be a frontier edge that has the smallest P-value
Add edge e to T, and set dist[y] = P(e)
b
7 (7)
a (0)
3 (3)
e
2 (2)
d (2)
k (8)
7
2
f
4
3
6
2
3
i
l (9)
c
8
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G
Update frontier edges
For each frontier edge e


•
•
let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
Let e be a frontier edge that has the smallest P-value
Add edge e to T, and set dist[y] = P(e)


a (0)
2 (2)
d (2)
3 (3)
e
k
i
7 (9) l
c
8
2
3
2
f
4
3
6 (8)
b
7 (7)
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G
Update frontier edges
For each frontier edge e


•
•
let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
Let e be a frontier edge that has the smallest P-value
Add edge e to T, and set dist[y] = P(e)


a (0)
7 (7)
2 (2)
3 (3)
e (3)
d (2)
k
i
7 (9) l
c
8
2
3
2
f
4
3
6 (8)
b
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G
Update frontier edges
For each frontier edge e


•
•
let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
Let e be a frontier edge that has the smallest P-value
Add edge e to T, and set dist[y] = P(e)


a (0)
7 (7)
2 (2)
3 (3)
e (3)
d (2)
k
i
7 (9) l
c
8
2
3 (6)
f
2
4 (7)
3 (6)
6 (8)
b
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G
Update frontier edges
For each frontier edge e


•
•
let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
Let e be a frontier edge that has the smallest P-value
Add edge e to T, and set dist[y] = P(e)


a (0)
7 (7)
2 (2)
3 (3)
e (3)
d (2)
3 (6)
6 (8)
k
i (6)
7 (9) l
b
c
8
2
3 (6)
f
2
4 (7)
g
2
4
j
h
4
9
9
11
m
Dijkstra's Shortest path finding algorithm

While T does not span G
Update frontier edges
For each frontier edge e


•
•
let x be the labelled and y the unlabelled endpoints of e
set P(e) = dist[x] + w(e)
Let e be a frontier edge that has the smallest P-value
Add edge e to T, and set dist[y] = P(e)


a (0)
7 (7)
2 (2)
3 (3)
e (3)
d (2)
3 (6)
6 (8)
k
i (6)
7 (9) l
b (6)
c
8
2
3 (6)
f
2
4 (7)
g
2
4
j
h
4
9
9
11
m
Graph theory
Topological properties of graphs
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Graph types

Homogeneous networks


Each pair of nodes is connected with a constant random probability
The connectivity follows a Poisson law
•
•
•


P(k) ~ lx*e-k/x!
l
mean number of connections per node
k
number of connections for a given node
The probability of finding a highly connected node decreases
exponentially with connectivity
Scale-free networks



The network contains a few highly connected nodes, and most other
nodes are weakly connected
Can be generated randomly with a model where new nodes are
preferentially connected to already established nodes
The connectivity follows a power law
•
•
•
P(k) ~ k-g
g
k
number of connections for a given node
Suggested readings

Gross, J. & Yellen, J. (1999). Graph theory and its
applications. Discrete mathematics and its applications
(Rosen, K. H., Ed.), CRC press, London
Basic concepts of graph theory
Exercises
Jacques van Helden
Jacques.van.Helden@ulb.ac.be
Adjacency matrix
a.
b.
c.
d.
Draw a graph on the basis of the adjacency matrix below.
Knowing that each cell of the matrix indicates the weight of an arc, label
the arcs on the graph.
Apply Dijkstra algorithm to find the shortest path from A to D in the graph.
(note: for the exam, another adjacency matrix will be used)
from\to
A
B
C
D
E
F
G
H
A
0
4
2
0
0
0
0
0
B
0
0
0
9
10
0
0
0
C
4
0
0
0
8
0
0
0
D
0
8
0
0
0
0
1
0
E
3
0
0
0
0
0
8
0
F
0
0
4
0
0
3
0
0
G
0
0
0
0
0
0
0
3
H
0
0
0
0
9
2
0
2
Answer: adjacency matrix
D
B
8
A
4
3
9
10
1
4
E
8
2
C
8
4
9
3
3
G
H
2
2
F
Tree traversal
Vertex
A
B
C
D
E
F
G
H
I
J
K
L
M
out
(E,D)
()
()
(I,H)
(B,G)
()
(C,F)
()
(K,M)
()
(L,J)
()
()
a.
b.
c.
Draw a graph corresponding to the adjacency list
below.
Which type of graph does it represent ?
Label the nodes according to the pre-order
traversal algorithm below. Indicate at each step
the state of the stack and the output.


Push root onto the stack
While stack is not empty
• Pop a vertex off stack, and write it to the output
list
• Push its children right-to-left onto stack
Tree traversal - answer
root = A
E
D
B
G
I
H
C
F
K
M
L
J
Exam March 2002 - question 1
a.
b.
Draw a graph on the basis of the adjacency matrix below.
Knowing that each cell of the matrix indicates the weight of an arc, label the arcs
on the graph.
Apply Dijkstra algorithm to find the shortest path from A to H in the graph.
c.
from\to
A
B
C
D
E
F
G
H
A B
C
D
E
F
G H
0
0
0
0
0
0
0
0
5
3
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
8
2
0
0
0
0
0
0
0
5
0
1
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
9
3
1
5
0
A
1
3
B
8
5
5
1
3
2
C
2
E
0
D
9
F
1
3
G
5
H
1
3
1
8
5
5,4
2
5
1
9,6
3
2
2
9
9,7
1
3
5
5
11,10,9,8
Exam March 2002 - question 2
a.
b.
c.
Draw the tree corresponding to the adjacency list below.
Which node is the root ?
Label the nodes according to the pre-order traversal algorithm
given below. Indicate at each step the state of the stack and the
output.


Vertex
A
B
C
D
E
F
G
H
I
J
K
Push root onto the stack
While stack is not empty
•
Pop a vertex off stack, and write it to the output list
•
Push its children right-to-left onto stack
out
(K,G)
()
(D,A)
()
(C,F)
()
(H)
()
()
()
(J,B,I)
E
C
F
D
A
K
J
G
B
I
H
Step
0
1
2
3
4
5
6
7
8
9
10
11
stack
out
E
F,C
E
F,A,D
C
F,A
D
F,G,K
A
F,G,I,B,J K
F,G,I,B
J
F,G,I
B
F,G
I
F,H
G
F
H
F
1
2
11
3
4
5
6
9
7
8
10
Exam June 2002 - question 1
a.
b.
Draw a graph on the basis of the adjacency matrix below.
Knowing that each cell of the matrix indicates the weight of an arc, label the arcs
on the graph.
Apply Dijkstra algorithm to find the shortest path from A to H in the graph.
c.
from\to
A
B
C
D
E
F
G
H
A B
C
D
E
F
G H
0
0
0
0
0
0
0
0
5
3
0
0
0
0
0
0
7
0
2
0
0
0
0
0
0
8
0
0
0
1
0
0
0
0
5
0
0
0
1
3
0
0
0
1
0
0
0
0
1
0
0
0
0
4
0
0
0
0
0
0
3
5
5
0
A
1
5
3
B
8
C
4
1
E
3
7
2
D
5
F
5 3
H
1
1
5
G
Exam June 2002 - question 2
a.
b.
c.
Draw the tree corresponding to the adjacency list below.
Which node is the root ?
Label the nodes according to the pre-order traversal algorithm
given below. Indicate at each step the state of the stack and the
output.


Vertex
A
B
C
D
E
F
G
H
I
J
K
Push root onto the stack
While stack is not empty
•
Pop a vertex off stack, and write it to the output list
•
Push its children right-to-left onto stack
out
(G)
()
(D,A,K)
()
(C,F,A)
()
(H)
(B,I)
()
()
(J)
E
C
D
F
A
K
G
J
H
B
I
Exam June 2002 - question 3
a.
b.
c.
Draw the tree corresponding to the adjacency list below.
Which node is the root ?
Label the nodes according to the level-order traversal algorithm
given below. Indicate at each step the state of the queue and the
output.


Enqueue root
While queue is not empty


Vertex
A
B
C
D
E
F
G
H
I
J
K
Dequeue a vertex and write it to the output list
Enqueue its children left-to-right
out
(G,E)
()
(D,A,K)
()
(C,F)
()
(H)
(B,I)
()
()
(J)
A
G
E
C
H
F
B
D
K
J
I
Exam June 2002 - question 4
a.
b.
c.
Draw an example of path
Draw an example of trail which is not a path
Draw an example of walk which is not a trail
Exam September 2002 - question 1
a.
b.
Draw a graph on the basis of the adjacency matrix below.
Knowing that each cell of the matrix indicates the weight of an arc, label the arcs
on the graph.
Apply Dijkstra algorithm to find the shortest path from A to H in the graph.
c.
A
from\to
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
1
H
1 5
5
1
B
2 4
1
0
1
C
5
1
1
5
4
1
1 1
10
2
4
1
4
2
4
1
4
10 7
1
2
2
D
1
1
10
H
7
E
10
1
2
G
2
F
3
1
1
10
12
7
10
1
2
9
4
2
11
Exam September 2002 - question 2
a.
b.
c.
Draw the tree corresponding to the adjacency list below.
Which node is the root ?
Label the nodes according to the level-order traversal algorithm
given below. Indicate at each step the state of the queue and the
output.


Enqueue root
While queue is not empty


Vertex
A
B
C
D
E
F
G
H
I
J
K
out
(B,C)
(D,E,F)
(G)
()
()
()
(H,I)
()
(J,K)
()
()
Dequeue a vertex and write it to the output list
Enqueue its children left-to-right
A
C
B
D
Step out queue
0
A
1
A B,C
2
B C,D,E,F
3
C D,E,F,G
4
D E,F,G
5
E F,G
6
F G
7
G H,I
8
H I
9
I J,K
10
J K
E
F
G
I
H
11
J
K
1
3
2
4
5
6
7
9
8
K
10
11
Exam July 2003 - question 1
a.
b.
Draw a graph on the basis of the adjacency matrix below.
Knowing that each cell of the matrix indicates the weight of an arc, label the arcs
on the graph.
Apply Dijkstra algorithm to find the shortest path from A to B in the graph.
c.
from\to
A
B
C
D
E
F
G
H
A
B
C
D
E
F
9
G
A
H
1 5
4
1 (1)
1
G (1)
3
5 (5)
3 (4)
9 (9)
H (4)
4 (8)
D (8)
1
15 (16) 11 (15)
1
15
10 4 9 11
A
3
1
G
15
F
5
11
10
C
1
1
F (13)
9
H
3
4
B
9 (13)
3 (11)
10 (14)
4
D
9
3
1
E
1 (13)
1 (14)
C (12)
1 (12)
4 (16)
B (14)
E (11)
Exam July 2003 - question 2
a.
b.
c.
Draw the tree corresponding to the adjacency list below.
Which node is the root ?
Label the nodes according to the pre-order traversal algorithm
given below. Indicate at each step the state of the stack and the
output.
Push root onto the stack
While stack is not empty
•
Pop a vertex off stack, and write it to the output list
•
Push its children right-to-left onto stack


Vertex
A
B
C
D
E
F
G
H
I
J
K
out
(I,N,H)
()
()
(J)
(P,G)
()
()
()
(D)
()
()
M
(K)
N
A
N
I
M
D
H
E
Step
0
1
2
3
4
5
6
7
8
9
10
stack
out
A
H,N,I
A
H,N,D
I
H,N,J
D
H,N
J
H,E,M
N
H,E,K
M
H,E
K
H,G,P
E
H,G,C,Q
P
H,G,C,F,B Q
11
H,G,C,F
B
12
H,G,C
F
13
H,G
C
(M,E)
14
H
G
P
(Q,C)
15
Q
(B,F)
J
K
P
Q
B
G
C
F
H
1
5
2
6
3
4
7
15
8
9 14
10 13
11
12