Introduction to Graph “theory”
• Why do we care about graph theory in testing
and quality analysis?
– The “flow” (both control and data) of a design,
within a program or among programs may be
depicted with a graphical representation.
• Graph gives us the tool to formally study
these graphical representations of flow and
design test cases from the flow.
What is a graph ?
•
•
A branch of topology—focus on connections
(undirected) Graphs
–
–
–
–
•
nodes, edges, matrices
degree of a node
Paths
Components
Directed graphs
–
–
–
–
–
–
–
nodes, edges
indegree, outdegree
paths and semi-paths
n-connectedness
cyclomatic number
strong and weak components
Program Graphs
What is a graph ?
n1
e1
e2
n3
n2
e3
n7
e4
n5
e5
n4
n6
A graph, G = { V, E }, is composed of :
i) a set of nodes, N, and
ii) a set of edges, E, which connects the nodes.
For the above example, V = { n1, n2, n3, n4, n5, n6, n7 } and
E = {e1, e2, e3, e4, e5 }
Degree of a node
• Note that in the previous graph, not every
node is immediately connected to another
node; and some node is immediately
connected to more than one other node.
– For example:
• N7 is not connected to any other node
• N3 is connected to three other nodes: n1, n4 and n5.
• Formally, the “degree” of a node x, nx, is
defined as the number of edges, e’s, that have
node nx as the immediate end point.
– This is the same as the number of other nodes that
nx is immediately connected to
• Degree (n7) = 0
• Degree (n3) = 3
For clarity, the definition uses edges, rather than immediately connect nodes.
Degree of a Node
n1
e1
n2
e4
n5
e2
n3
e3
n4
e5
n6
n7
deg(n1) = 2
deg(n2) = 2
deg(n3) = 1
deg(n4) = 3
deg(n5) = 1
deg(n6) = 1
deg(n7) = 0
Draw this on the borad!
n1
e1
e2
n3
n2
e3
n7
e4
n5
e5
n4
n6
A graph, G = { V, E }, is composed of :
i) a set of nodes, N, and
ii) a set of edges, E, which connects the nodes.
For the above example, V = { n1, n2, n3, n4, n5, n6, n7 } and
E = {e1, e2, e3, e4, e5 }
Using Incidence matrix to represent graph
e1 e2 e3 e4 e5
n1 1
1
n2 1
n3
n4
1
1
1
1
n5
1 1
n6
1
n7
An incidence matrix of a graph composed of J nodes
and K edges is a J x K cross-product relation, where
the rows are the nodes and the columns are the edges.
The relation, in this case, is “immediately connected.”
For this example, the incidence matrix represents
a 7 x 5 cross-product relation. The 1 in a cell, (x,y)
represents that node, nx, is the end point of edge, ey.
The empty cells may be filled with 0’s.
1)The sum of 1’s for any row, which represents a
node, is the degree of that node.
- e.g. sum of 1’s for node , n5, is 2; therefore
degree(n5) = 2
2)Each column shows the edge, ex, and the 1’s in that
column shows the nodes that edge, ex, connects
(note each column always adds up to 2.)
Sample Incidence Matrix
(What must be true about columns in an incidence matrix?)
e1 e2 e3 e4 e5
n1
1
1
0
0
0
n2
1
0
0
1
0
n3
0
0
1
0
0
n4
0
1
1
0
1
n5
0
0
0
1
0
n6
0
0
0
0
1
n7
0
0
0
0
0
n1
e1
n2
e4
n5
e2
n3
e3
n4
e5
n6
n7
Using Adjacency matrix to represent graph
An adjacency matrix of a graph composed of Z
n1 n2 n3 n4 n5 n6 n7 number of nodes is a Z x Z relation where the
rows are nodes and the columns are also nodes.
1 1
n1
Again the relation is “immediately connected.”
n2 1
n3 1
1
n4
1
n5
1
n6
n7
1
1
1
For this example, the adjacency matrix
represents a 7 x 7 cross-product relation.
The 1 in a cell, (x,y) represents that node, nx, is
immediately connected to another node, ny.
The empty cells may be filled with 0’s.
1)The sum of 1’s for any row or any column is
the degree of that node.
- e.g. sum of 1’s for node , n3, is 3; thus
degree(n3) = 3
2)Note that the matrix is symmetric across the
diagonal.
Sample Adjacency Matrix
n1
n2
n3
n4
n5
n6
n7
n1
0
1
0
1
0
0
0
n2
1
0
0
0
1
0
0
n3
0
0
0
1
0
0
0
n1
e1
n2
e4
n5
e2
n3
e3
n4
n4
1
0
1
0
0
1
0
n5
0
1
0
0
0
0
0
e5
n6
0
0
0
1
0
0
0
n6
n7
0
0
0
0
0
0
0
n7
Path
• Note that we used the term, “immediately connected”
for degree of a node.
• A path in a graph is a sequence of immediately
connected nodes: e.g. {nj, -- - , nj+z } in the graph
where nj is immediately connected to nj+1, and node
nj+1 is immediately connected to nj+2, and so on to
node nj+z.
• A path in a graph may also be defined as a sequence
of edges, such that for any pair of edges, ei and ej, in
the sequence, the edges share a common node.
Paths
n1
e1
e2
n3
n2
e3
n7
e4
n5
e5
n4
n6
There are several paths in the example graph.
node sequence to depict a path
Path1 : n2, n1, n3, n5
Path2 : n1, n3, n4
edge sequence to depict a path
e1, e2, e4
e2, e3
You may use either approach; I may mix and use both
Some Paths
path between
– n1 and n5
– n6 and n5
– n3 and n2
n1, n2, n5
n6, n4, n1, n2, n5
n3, n4, n1, n2
n1
e1
n2
e4
n5
e2
n3
e3
n4
e5
n6
n7
e1, e4
e5, e2, e1, e4
e3, e2, e1
Connected nodes (Connectedness)
• Earlier two nodes are said to be immediately
connected if there is an edge that connect
the two nodes. The degree of a node was
determined by the number of its immediately
connected nodes.
• Now a more general notion of connectedness
of two nodes may be given.
Node nx is connected to node ny if there is a
path from nx to ny
Properties of Connectedness
• Connected nodes on a path have some
interesting relational properties:
– It is reflexive because every node is connected to
itself and thus is on the same path.
– It is symmetric because nodes ni and nj on a path
implies that nodes nj and ni are also on the same
path.
– It is transitive for nodes ni and nj on a path and
nodes nj and nk on the same path implies that ni
and nk are on the same path also.
Note: This relation of ‘connected’ satisfies the definition of equivalence relation.
Component and condensation graph
• A component of a graph is defined as the maximal
set of connected nodes.
– From our graph example:
• {n1, n2, n3, n4, n5, n6} form a component
• {n7} also forms a component
• A condensation graph is formed by replacing each
component in the original graph by a “condensed”
single node. There are no edges.
– Condensation graph of a graph gives us a view of the
“complexity” through the number of components.
Condensed Graph
n1
e1
e2
n3
n2
e3
n7
e4
n5
e5
n4
n6
N1
N2
From design/coding/testing perspective, this says we have two major
“chunks” or components that we need to worry about.
Cyclomatic Number of a Graph
The cyclomatic number of a graph G is given by V(G) = e – n +
p, where
– e is the number of edges in G
– n is the number of nodes in G
– p is the number of components in G
• Cyclomatic complexity pertains to both ordinary and directed
graphs (next topic).
•The graph must be strongly connected.
• not very useful on our running example because it is not
strongly connected. We will see this part in the future chapters.
Cyclomatic Complexity Example
n1
e1
n2
e4
n5
e2
n3
e3
n4
e5
n6
n7
V(G) = e – n + p
=5–7+2
=0
Directed Graph
n1
e1
e2
n3
n2
e3
n7
e4
n5
e5
n4
n6
A directed graph, DG = { V, E }, is composed of :
i) a set of nodes, V, and
ii) a set of edges, E, which connects the nodes in “order”
(the set of edges have directions)
For example:
e1 does not only connect nodes n2 and n1, but shows that the order is n1 to n2.
Thus e1 may viewed as an ordered tuple, <n1, n2>, which is different from <n2, n1>.
However, (n1, n2) is an unordered pair.
Directed Graph (another example)
n1
e1
n2
e4
n5
e2
n3
e3
n4
n7
e5
n6
V = {n1, n2, n3, n4, n5, n6, n7}
E = {e1, e2, e3, e4, e5}
= {<n1, n2>, <n1, n4>, <n3, n4>, <n2, n5>, <n4, n6>}
Degrees of Directed Graph
•
With directed graph, one can utilize the
directional concept.
–
–
•
e.g. the flow of a program, the control flow in imperative
programming language, can be represented by directed
graph.
Each edge not only connects nodes but shows a
direction (like a vector).
For each node, n, in the graph, there may be
defined:
1. Indegree (n) = the number of distinct edges
immediately directed in and towards n.
2. Outdegree (n) = number of distinct edges
immediately directed out and away from n.
Note that for any node, n:
degree (n) = indegree (n) + outdegree (n)
Indegrees and Outdegrees
n1
indeg(n1) = 0
indeg(n2) = 1
indeg(n3) = 0
indeg(n4) = 2
indeg(n5) = 1
indeg(n6) = 1
indeg(n7) = 0
outdeg(n1) = 2
outdeg(n2) = 1
outdeg(n3) = 1
outdeg(n4) = 1
outdeg(n5) = 0
outdeg(n6) = 0
outdeg(n7) = 0
e1
n2
e4
n5
e2
n3
e3
n4
e5
n6
n7
Some “types” of node for directed graph
• Source node is a node with indegree = 0.
• Sink node is a node with outdegree = 0.
• Transfer node is node with indegree ≠ 0 and
outdegree ≠ 0.
n1
n4 is a source node.
n1 and n2 are transfer nodes.
n3 is a sink node.
n2
n4
n3
Adjacency Matrix for Directed Graph
‘to’
nodes
n1
n1
‘from’
nodes
n1
n2
n3
n4
1
1
n2
n3
n4
1
n2
n4
n3
1) There is a directed edge from n4 to n1.
Thus there is a 1 in cell (4,1).
2) There is a 1 in cell (1,2) to indicate that
there is a directed edge from n1 to n2.
3) And finally, the 1 in cell (2,3) represents
the directed edge from n2 to n3
Path and semi-path
• With directed graph, the notion of a path is not just
connection, but there is a directional connection.
– 1. A directed path is a sequence of edges such that for any
adjacent pair of edges, e1 and e2, the terminal node for e1 is
the starting node for e2.
– 2. A cycle is a directed path that begins and ends at the
same node
– 3. A chain is a sequence of nodes in which every interior
node has indegree = outdegree = 1
– 4. A directed semi-path is a sequence of edges such that,
for at least one adjacent pair of edges, ei and ej, in the
sequence, the initial node of the first edge is the same initial
node of the second edge or the terminating node of the first
edge is the same as the terminating node of the second
edge.
Path, cycle, and semi-path
1. There is a directed path from n1 to n3.
2. There is cycle “among” n1, n2, and n4.
n1
3. There is a directed semi-path between n3 and n4
n2
n4
n3
Note that we linked
n2 to n4 to create a
cycle
Directed Graph (new example)
n1
e1
n2
e4
n5
V = {n1, n2, n3, n4, n5, n6, n7}
E = {e1, e2, e3, e4, e5}
= {(n1, n2), (n1, n4), (n3, n4),
(n2, n5), (n4, n6)}
n7
There is a path from n1 to n6,
and there are semipaths between
nodes n1 and n3, n2 and n4, and
between nodes n5 and n6.
e2
n3
e3
n4
e5
n6
Reachability with Adjacent Matrix Operations
• Adjacent matrix for a directed graph indicates the
directional connection of the nodes, not a two way
connection. Note that an edge that connects nodes,
nx and ny does not mean it connects ny and nx in
the reverse way. So for directed graph the “connect”
operation is non-symmetric.
• We can use matrix operation on Adjacent matrix of
the directed graph to view the paths in the directed
graph.
An example of operating on Adjacent matrix
for directed graph
n1
n1
A1 =
n1
n2
n3
n4
1
1
n2
n3
n4
1
n2
n4
n3
A2 =
0100
0010 X
0000
1000
0100
0010
0000
1000
=
0010
0000
0000
0100
Note that A1 gives connections length 1, and A2 gives the
connections of length 2 in the directed graph.
More on Adjacent Matrix operation of
A3=
0100
0010
0000
1000
X
0100
0010
0000
1000
X
0100
0010
0000
1000
=
0000
0000
0000
0010
A3 shows only one path, from n4 to n3, that is length 3 in the directed graph
A4=
0100
0010 X
0000
1000
0100
0010 X
0000
1000
0100
0010 X
0000
1000
0100
0010
0000
1000
=
A4 shows that there is no length 4 path in the directed graph
0000
0000
0000
0000
Reachability of directed graph using
Adjacent matrices
- Assuming that every node can reach itself, we will
not indicate the reachability of a node to itself.
-Reachability, R, of a directed graph may be derived from the
adjacency matrix, A, of that directed graph as follows:
R = A1 + A2 + - - - - + Ak, where Ak is non-zero, but Ak+1 is zero
n1
R=
A1
+
A2
+
A3
1 23 4
n2
n4
R=
n3
0100
0010
0000
0010 + 0000 + 0000
0000
0000
0000 =
1000
0100
0010
1
2
3
4
0110
0010
0000
1110
If you were interested in a variable or a constant that is defined in one
node and where it could “possibly” be used ---- this may help
Reachability Matrix
n1
n2
n3
n4
n5
n6
n7
n1
1
1
0
1
1
1
0
n2
0
1
0
0
1
0
0
n3
0
0
1
1
0
1
0
n4
0
0
0
1
0
1
0
n5
0
0
0
0
1
0
0
n6
0
0
0
0
0
1
0
n7
0
0
0
0
0
0
1
n1
e1
n2
e4
n5
e2
n3
e3
n4
e5
n6
n7
Compute A2
n1
n2
n3
n4
n5
n6
n7
n1
n2
n3
n4
n5
n6
n7
n1
0
1
0
1
0
0
0
n1
0
1
0
1
0
0
0
n2
0
0
0
0
1
0
0
n2
0
0
0
0
1
0
0
n3
0
0
0
1
0
0
0
n3
0
0
0
1
0
0
0
n4
0
0
0
0
0
1
0
n4
0
0
0
0
0
1
0
n5
0
0
0
0
0
0
0
n5
0
0
0
0
0
0
0
n6
0
0
0
0
0
0
0
n6
0
0
0
0
0
0
0
n7
0
0
0
0
0
0
0
n7
0
0
0
0
0
0
0
n1
n2
n3
n4
n5
n6
n7
n1
e1
n2
e4
n5
n1
n2
e2
n3
n4
n5
n3
e3
n4
n6
n7
e5
n6
n7
n-Connectedness
Nodes nj and nk in a directed graph are
– 0-connected iff no path (or semipath) exists between nj and nk
– 1-connected iff a semi-path but no path exists between nj and
nk
– 2-connected iff a path exists from nj and nk
– 3-connected iff a path goes from nj to nk and a path goes from
nk to nj
• No other degrees of n-connectedness exist
• A new edge, e6, is added to our continuing example.
• n-connectedness has very useful expressive power.
Directed Graph (third version)
n1
e1
n2
e4
n5
e2
n3
e3
e6
n4
e5
n6
n7
Directed Graph (third version)
n1
e1
n2
e4
n5
e2
n3
e3
e6
n4
n7
e5
n6
Find examples of a chain, a cycle, a set
of 3-connected nodes
Strong Components of a Directed Graph
A strong component of a directed graph is a
maximal set of 3-connected nodes.
Strong components...
– identify loops and isolated nodes.
– lead to another form of condensation graph
– support an excellent view of testing programs with loops
Directed Graph (third version)
n1
e1
n2
e4
n5
e2
n3
e3
e6
n4
n7
e5
n6
Strong components: S1 = {n3, n4, n6}, S2 = {n7}
Condensation Graph of a Directed Graph
Given a directed graph D = (V, E), its condensation
graph is formed by replacing strongly connected
nodes by their corresponding strong components.
A condensation graph of a directed graph...
– contains no loops, and is therefore
– a Directed Acyclic Graph (DAG)
– support an excellent view of testing programs with loops
Condensation Graph of Directed Graph
(third version)
n1
e1
n2
e4
n5
e2
n3
e3
e6
n1
e1
n2
e4
n5
e2
n4
n7
S1
e5
n6
Strong components: S1 = {n3, n4, n6}, S2 = {n7}
S2
Program Graphs
The program graph of a program written in an
imperative programming language is a directed
graph in which nodes are either entire statements or
statement fragments. There is an edge from node i
to node j if and only if node j can be executed
immediately after node i).
• (The early (1970s) definitions referred to nodes as
entire statements, but this doesn’t fit well with
modern programming languages.)
• We shall use “statement fragment” to refer either
to full statements or to statement fragments.
Program Graphs
Program Graphs
• When drawing a program graph, it is usually
simpler to number the statement fragments.
• This Figure
If-Then-Else
1
2
3
4
5
6
7
If <condition>
Then
<then statements>
Else
<else statements>
End If
<next statement>
Pre-test Loop
1
1
2
3
4
4
2
3
5
While <condition>
<repeated body>
End While
<next statement>
1
2
3
6
4
7
Case/Switch
1
2
3
4
5
6
7
8
Case n Of 3
n=1:
<case 1 statements>
n=2:
<case 2 statements>
n=3:
<case 3 statements>
End Case
1
2
3
4
5
8
Post-test Loop
6
7
1
2
3
4
Do
<repeated body>
Until <condition>
<next statement>
1
2
3
4
Four Graph-Based Models
•
•
•
•
Finite State Machines
Petri Nets
Event-Driven Petri Nets
StateCharts
• These are all executable models, i.e., it is
possible to build a program (an engine) to execute
the model.
Finite State Machines (FSMs)
• Finite State Machines are directed graphs in which nodes
are states, and edges are transitions from one state to a
successor state.
• Transitions are caused by
– events
– date conditions
– passage of time (an event)
• Constraints
– states are mutually exclusive
– only one transition can occur at a time
• FSMs are ideally suited for menu-driven applications
Garage Door Controller FSM
Input events
e1: depress controller button
e2: end of down track hit
e3: end of up track hit
e4: obstacle hit
e5: laser beam crossed
Output events (actions)
a1: start drive motor down
a2: start drive motor up
a3: stop drive motor
a4. door stops part way
a5. door continues opening
a6. door continues closing
s1:
Door Up
e1/a1
e3/a3
s5:
Door
Closing
e1/a3
e4/a3, a2
e1/a1
s3:
Door Stopped
Going down
s4:
Door Stopped
Going Up
e1/a2
e5/a3, a2
e2/a3
s6:
Door
Opening
e1/a2
s2:
Door down
e1/a3
Graph Representations
Will need these later:
• Program Control Flows: sequential, if-then-else, loop,
etc.
• State Transition Diagram (Finite State Machines):
states and stimuli
Will not use:
• Petri Net: a complex set of events and transitions
• State Charts: A complicated (embedding of states)
form of State Transition Diagram