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J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 13: Graphs, part II.
Page 2
e
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
induced by
is not connected.
= subgraph of
J. Maluszynski, IDA, Linköpings Universitet, 2004.
[L/D p. 440–442]
and has more than one child in , or
f 3
TDDB57 DALG – Lecture 13: Graphs, part II.
4
no skipped edge joins a descendant
of f with a proper ancestor of b
cut vertex
Biconnectivity
c
e 5
Consider fault-tolerance in a communication network.
One node breaks down, can the other ones still communicate?
b
d
Overview
[L/D 12.2]
c
2 b
Some applications of DFS
a
e
Page 4
is a cutvertex of
it is not a root and
it has a child in , none of whose descendants
is connected with a proper ancestor of by a skipped edge.
1 a
is a cutvertex of a connected graph
b
d
cutvertex: a
A graph is biconnected if it has no cutvertices.
? How to check this?
TDDB57 DALG – Lecture 13: Graphs, part II.
it is the root of DFS tree
Checking biconnectivity of undirected graph
Topological sorting of directed acyclic graphs
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Depth-First Search Tree and Cut Vertices
Page 3
Depth-First Search Tree
f 3
Visit:
a
b
f
c
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Encountered:
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a,b
a,b,f
a,b,f,c
a,b,f,c,e
a,b,f,c,e,d
a,b,f,c,e,d
6 d
TDDB57 DALG – Lecture 13: Graphs, part II.
Lemma:
2 b
c 4
e 5
d
Depth-First Search follows some edges by its recursive calls
and skips the other (target vertex already visited)
1 a
6 d
Pre-visit version of DFS
The vertices of with the followed edges make a tree . Why?
These followed edges are called tree edges, the other skipped edges.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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Page 8
Checking Biconnectivity (2)
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[L/D Algorithm 12.4]
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89:
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Page 7
TDDB57 DALG – Lecture 13: Graphs, part II.
Topological Sorting of a DAG: Implementation with DFS
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Checking Biconnectivity (1)
Topological Sorting of a Directed Acyclic Graph (DAG)
Find cutvertices of
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We could use depth-first search, where
Topological sorting:
Given a DAG , put its vertices in a sequence
such that if
is an edge of then
.
i.e., vertices in a linear order preserving the “arrow” ordering.
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TDDB57 DALG – Lecture 13: Graphs, part II.
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TDDB57 DALG – Lecture 13: Graphs, part II.
Adjusting DFS to Topological Sort of DAGs
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J. Maluszynski, IDA, Linköpings Universitet, 2004.