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End-vertices of graph searching algorithms
Graph
searching
algorithms
Properties
of endvertcies
End-vertex
Problems
Shou-Jun Xu (MÅ )
E-mail:[email protected]
Lanzhou University
@ Shanghai Jiao Tong University, China
Jun 6, 2014
Outlines
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Graph
searching
algorithms
1
Graph searching algorithms
2
Properties of end-vertcies
3
End-vertex Problems
Properties
of endvertcies
End-vertex
Problems
Graph searching algorithms
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Graph
searching
algorithms
LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
• General graph searching algorithm: employing some mechanism
for systematically visiting all vertices and edges in the given graph.
After choosing an initial vertex, a search of a connected graph visits
each of the vertices and edges of the graph such that a new vertex
is visited only if it is adjacent to some previously visited vertex.
• Several well-known graph searching algorithms
BFS (Breadth-first search);
DFS (Depth-first search);
LBFS (Lexicographic BFS, Rose et al., 1976);
LDFS (Lexicographic DFS, Corneil et al., 2008);
MCS (Maximum cardinality searching, Tarjan et al., 1984);
MNS (Maximal neighborhood searching, Corneil et al., 2008).
LBFS
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(The following algorithm cut from Reference [Corneil et al., 2008])
Graph
searching
algorithms
LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
D.J. Rose, R.E. Tarjan, G.S. Lueker, Algorithmic aspects of vertex elimination on
graphs, SIAM J. Comput. 5 (1976) 266–283.
An example for LBFS
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Graph
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LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
{5-1}
1
{6}
{4}
{5-1}
{5-1}
1
{6}
{4, 5-2 }
2 {4}
( A)
{4, 5-3 }
{ 5-2 }
3
{4, 3}
2 {4}
{ 3}
2 {4} ( C )
( B)
4 {4, 2}
1
{6}
3
{4, 3}
1
{6}
4 {4, 2}
3
{4, 3}
1
{6}
{ 3, 5-4 }
( D)
2 {4}
5 { 3, 1}
( E)
LexBFS
• Note that at any step, if we denote l(u) = (a1 , a2 , · · · , ak ), then
ai is the number of visited neighbours of u, l(u) is a sequence in
the decreasing order and the order of visited neighors of u is a1
first, a2 second, and so on.
LDFS
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(The following algorithm cut from Reference [Corneil et al., 2008])
Graph
searching
algorithms
LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
D.G. Corneil, R.M. Krueger, A unified view of graph searching, SIAM J. Discrete
Math. 22 (2008) 1259–1276.
An example for LDFS
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Graph
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LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
{1}
1
{0}
{ 2 ,1}
1
{0}
{1}
{1}
2 {1}
{ 2 ,1} ( B )
( A)
3 {2, 1}
1
{0}
2 {1}
{ 4 , 2, 1}
3 {2, 1}
( D)
4 {3}
LexDFS
1
{0}
1
{0}
2 {1}
{3}
{2, 1}
3 {2, 1}
( C)
2 {1}
5 {4, 2, 1}
4 {3}
( E)
• Note that at any step, if we denote l(u) = (a1 , a2 , · · · , ak ), then
ai is the number of visited neighbours of u, l(u) is a sequence in
the decreasing order and the order of visited neighors of u is ak
first, ak−1 second, and so on.
MCS
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(The following algorithm cut from Reference [Berry et al., 2010])
Shou-Jun
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Graph
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LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
R.E. Tarjan, M. Yannakakis, Simple linear-time algorithms to test chodality of graphs,
test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J.
Comput. 13 (1984) 566–579.
An exmaple for MCS
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Graph
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LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
1
2
1
0
1
0
1
1
( A)
1
0
2 1
2
End-vertex
Problems
4 2
( D)
1
0
2 1
2
3 2
3 2
MCS
( B)
2
3 2
1
0
2 1
4 2
2 1
( E)
( C)
5 2
1
MNS
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(The following algorithm cut from Reference [Corneil et al., 2008])
Graph
searching
algorithms
LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
D.G. Corneil, R.M. krueger, A unified view of graph searching, SIAM J. Discrete
Math. 22 (2008) 1259–1276.
Notes on algorithms
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Graph
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LBFS
LDFS
MCS
MNS
notes on
algorithms
Properties
of endvertcies
End-vertex
Problems
• Let G be a connected graph with vertex set V of size n. We
define α = (α(1), α(2), · · · , α(n)) to be a linear ordering of V ,
which will typically be the left-to-right order in which we number
the vertices in a search, the smaller the number is labeled, the
earlier the vertex is visited.
Some basic definitions
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Graph
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Properties
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Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
Let G be a connected graph. A vertex v in G is called an
end-vertex of some graph searching algorithm (such as LBFS,
LDFS), if v is visited last by some execution of the algorithm
on G.
A chordal graph is a graph with no chordless cycle C of length
greater than 3, i.e., there is an edge connecting nonconsecutive vertices on any cycle of length greater than 3.
A clique is a set of pairwise adjacent vertices.
A vertex is simplicial if its neighborhood is a clique.
A path P misses vertex v (or v misses P ) if P ∩ N (v) = ∅
(i.e., no vertex of P is adjacent to v).
Two vertices x, y are unrelated with respect to vertex v if
there are paths P between x and v and Q between y and v
such that P misses y and Q misses x.
An independent triple of vertices x, y, z is an Asteroidal triple
(AT), if between every pair of vertices, there is a path that
misses the third.
Applications of end-verties of LBFS
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Graph
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Properties
of endvertcies
Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
(Dragan et al., 1997) The eccentricity ecc(v) of an end-vertex
v is equal to the graph’s diameter diam(G) if G is interval.
ecc(v) > diam(G) − 1 if G is HHD-free (Dragan, 1999),
chordal (Dragan et al. 1997) or AT-free (Corneil, 2001).
ecc(v) > diam(G) − 2 if G is HH-free (Dragan, 1999).
For these restricted families of graphs, there are an easy, linear
time algorithm that closely approximates the diameter of the
graph.
(Corneil et al, 1999) End-vertices can be used to find a dominating pair in connected AT-free graph G.
F.F. Dragan, F. Nicolai, A. Brandstadt, LDFS-orderings and powers of graphs, in:
LNCS 1197 (1997) 166–180.
F.F. Dragan, Almost diameter of a house-hole-free graph in linear time via LBFS,
Discrete Appl. Math. 95 (1999) 223-239.
D.G. Corneil, F.F. Dragan, M. Habib, C. Paul, Diameter determination on restricted
graph families, Discrete Appl. Math. 113 (2001) 143-166.
Basic properties of end-vertices
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Graph
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Properties
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Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
(Rose et al., 1976) The end-vertex of an LBFS of a chordal
graph is simplicial.
(Tarjan, Yannakakis, 1984) The end-vertex of an MCS of a
chordal graph is simplicial.
(Corneil et al., 2008) The end vertex of an MNS (say an
LDFS) of a chordal graph is simplicial.
(Corneil et al., 1999) The end-vertex of an LBFS of an ATfree graph is admissible.
D.J. Rose, R.E. Tarjan, G.S. Lueker, Algorithmic aspects of vertex elimination on
graphs, SIAM J. Comput. 5 (1976) 266–283.
R.E. Tarjan, M. Yannakakis, Simple linear-time algorithms to test chodality of graphs,
test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J.
Comput. 13 (1984) 566–579.
D.G. Corneil, R.M. krueger, A unified view of graph searching, SIAM J. Discrete
Math. 22 (2008) 1259–1276.
D.G. Corneil, S. Olariu, L. Stewart, Linear time algorithms for dominating pairs in
asteroidal triple-free graphs, SIAM J. Comput 28 (1999) 1284–1297.
Basic definitions about Moplex
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Graph
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Properties
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Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
A subset S of V is called a separator if the graph obtained by
deleting S together with the edges connected with S from G
is disconnected. This defines a set of connected components
of S.
A separator S is called an ab-separator if vertices a and b lie
in two different components of S.
S is called a minimal ab-separator if S is an ab-separator and
no proper subset of S is also an ab-separator.
S is called a minimal separator if there is some pair of vertices
a and b such that S is a minimal ab-separator. Alternatively,
S is a minimal separator if and only if there are at least two
connected component C1 and C2 of S such that N (C1 ) =
N (C2 ) = S; such components are called full components of
S.
A module is a subset A of V such that the vertices of A share
the same external neighborhood.
Illustrations for definitions, such as, moplex
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• A subset A of V is called a maximal clique module if A is
both a clique and a module and A is inclusion-maximal for both
properties.
• A moplex X is both a clique and a module such that N (X) is
a minimal separator.
• (Xu et al., 2013) A moplex is a maximal clique module.
Properties
of endvertcies
Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
S1
V2
V1
V3
V4
V5
V7
S2
V6
V8
V9
S1 is a moplex and S2 is a maximal clique module but not a moplex
in the graph G. minimal separators: {v6 }, {v5 , v6 }, {v7 , v8 }.
Notes on moplex
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Properties
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Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
The vertices of a moplex are in some sense equivalent, because
they share the same external neighborhood.
The notion of moplex strenghthens the notion of simplicial
vertex, as in a chordal graph any vertex of a moplex is simplicial, whereas in some chordal graphs, there may be simplicial
vertices which do not belong to any moplex.
For example, a is simplicial but does not belong to a moplex.
(The following figure cut from Reference [Berry et al., 2010])
Berry et al. results on moplex
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Properties
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Some basic
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Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
LBFS on a non-clique graph ends on a vertex which belongs
to a moplex.
In an arbitrary graph, The moplex to which the end-vertex by
some execution of LBFS belongs are numbered consecutively.
For any graph and an end-vertex v of LBFS, the minimal
separators included in N (v) are totally ordered by inclusion.
For any graph G, any LBFS ordering α of G numbers the
thin slices in order.
D.G. Corneil, E. Kohler, J.-M. Lanlignel, On end-vertices of Lexicographic Breadth
First Searches, Discrete Appl. Math., 158 (2010) 434-443.
A. Berry, J.R.S. Blair, J.-P. Bordat, G. Simonet, Graph Extremities Defined by Search
Algorithms, Algorithms, 3 (2010) 100–124.
Illustration for Berry et al. rusults
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Graph
searching
algorithms
Properties
of endvertcies
Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
The moplex {10, 9}; minimal separators included in N (10): {8, 9}(⊂
){3, 4, 8, 9}; thin slices: {1, 2, 3, 4}, {5}, {6, 7, 8}.
Xu et al. results on moplex
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Graph
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Properties
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Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
LDFS on a non-clique graph ends on a vertex which belongs
to a moplex.
In an arbitrary graph, The moplex to which the end-vertex by
some execution of LDFS belongs are numbered consecutively.
For any graph G and an end-vertex v of LBFS, are the minimal separators included in N (v) totally ordered by inclusion?
WRONG
For any graph G, is any LBFS ordering α of G number the
thin slices in order? WRONG
S.-J. Xu, X. Li, R. Liang, Moplex orderings generated by the LDFS algorithm, Discrete
Appl. Math. 161 (2013) 2189–2195.
Counterexamples
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Graph
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Properties
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Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
5
1
7
4
3
2
(A)
6
9
3
8
6
4
2
5
1
(B)
(A) The moplex {9}; N (9) = {1, 3, 4, 6, 8}; minimal separators
included in N (9): {1, 3, 4, 8}, {1, 3, 4, 6}, {1, 3, 4, 6, 8}, there are
no inclusion relations;
(B) thin slices: {1, 2, 4, 5}, {3}.
Moplex property of end-vertices of MCS
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Graph
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MCS on a chordal graph ends on a vertex which belongs to
a moplex.
On a non-chordal graph?? WRONG
Properties
of endvertcies
Some basic
definitions
Applications
of
end-vertices
of LBFS
Properties of
end-vertices
Basic
properties of
end-vertices
Moplex
property of
end-vertices
Moplex
property of
End-vertices
of LBFS
Moplex
properties of
end-vertices
of LDFS
End-vertex
6
4
5
2
1
3
The end-vertex 6 is not a vertex of a moplex.
Characterizations of end-vertices
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Graph
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Properties
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End-vertex
Problems
Characterizations
of
end-vertices
Complexity
of end-vertex
Problems
(Corneil et al., 2009) A vertex of an interval graph is an endvertex of an LBFS if and only if it is both simplicial and
admissible.
(Corneil et al., 2010) Let G be an arbitrary graph. If x is a
simplical and admissible vertex of G, then there is an LBFS
of G ending at x.
(Corneil et al., 2009) A vertex t of a chordal graph is an
end-vertex of MNS if and only if t is simplicial and the minimal separators of G included in N (t) are totally ordered by
inclusion.
(Charbit et al., 2014) A vertex t is an end-vertex of generic
graph search of graph G if and only if t is not 1-cutset of G.
D.G. Corneil, S. Olariu, L. Stewart, The LBFS structure and recognition of interval
graphs, SIAM J. Discrete Mathematics, 23 (2009) 1905õ1953
D.G. Corneil, E. Kohler, J.-M. Lanlignel, On end-vertices of Lexicographic Breadth
First Searches, Discrete Appl. Math., 158 (2010) 434-443.
P. Charbit, M. Habib, A. Mamcarz, Influence of the tie-break rule on the end-vertex
Descriptions for End-vertex Problems
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Graph
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Properties
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End-vertex
Problems
Characterizations
of
end-vertices
Complexity
of end-vertex
Problems
(The following problems cut from Reference [Charbit et al., 2014])
Complexity of end-vertex problem
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Graph
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Properties
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End-vertex
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Characterizations
of
end-vertices
Complexity
of end-vertex
Problems
(Charbit et al., 2014) The end-vertex and the beginning-endvertex problems for the generic search can be solved in linear
time.
(Corneil et al. 2010) The end-vertex problem of LBFS for
interval graphs can be computed in the linear time.
(Berry et al., 2010) The end-vertex problem of MNS is polynomial for the class of chordal graphs.
D.G. Corneil, E. Kohler, J.-M. Lanlignel, On end-vertices of Lexicographic Breadth
First Searches, Discrete Appl. Math., 158 (2010) 434-443.
A. Berry, J.R.S. Blair, J.-P. Bordat, G. Simonet, Graph Extremities Defined by Search
Algorithms, Algorithms, 3 (2010) 100–124.
P. Charbit, M. Habib, A. Mamcarz, Influence of the tie-break rule on the end-vertex
problem, http://www.liafa.univ-paris-diderot.fr/ mamcarz/papers/endvertices.pdf
A summary of Carbit et al. results on complexity
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(The following table cut from Reference [Charbit et al., 2014])
Graph
searching
algorithms
Properties
of endvertcies
End-vertex
Problems
Characterizations
of
end-vertices
Complexity
of end-vertex
Problems
P. Charbit, M. Habib, A. Mamcarz, Influence of the tie-break rule on the end-vertex
problem, http://www.liafa.univ-paris-diderot.fr/ mamcarz/papers/endvertices.pdf
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Graph
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Properties
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End-vertex
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Characterizations
of
end-vertices
Complexity
of end-vertex
Problems
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