End-vertices of graph searching algorithms Graph searching - - PowerPoint PPT Presentation
Report in Shanghai Shou-Jun Xu ( M ) End-vertices of graph searching algorithms Graph searching algorithms Shou-Jun Xu ( M ) Properties of end- vertcies E-mail:shjxu@lzu.edu.cn End-vertex Lanzhou University Problems @
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies End-vertex Problems
End-vertices of graph searching algorithms
Shou-Jun Xu (MÅ) E-mail:shjxu@lzu.edu.cn Lanzhou University @ Shanghai Jiao Tong University, China Jun 6, 2014
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies End-vertex Problems
Outlines
1
Graph searching algorithms
2
Properties of end-vertcies
3
End-vertex Problems
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
Graph searching algorithms
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.
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).
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
LBFS
(The following algorithm cut from Reference [Corneil et al., 2008]) D.J. Rose, R.E. Tarjan, G.S. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput. 5 (1976) 266–283.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
An example for LBFS
2 1 {6} 1 {4} {4, } 5-2 {5-1} {5-1} {5-1} 3 4 5 { } 5-2 {4} 2 1 {4} {4, 3} { 3} {4, } 5-3 3 2 1 {4} {4, 3} { 3, } 5-4 {4, 2} 4 3 2 1 {4} {4, 3} { 3, 1} {4, 2}
( ) A ( ) B ( ) C ( ) E ( ) D
LexBFS {6} {6} {6} {6}
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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
LDFS
(The following algorithm cut from Reference [Corneil et al., 2008]) D.G. Corneil, R.M. Krueger, A unified view of graph searching, SIAM J. Discrete
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
An example for LDFS
2 1 {0} 1 {0} {1} {1} {1} {1} 3 { ,1} 2 1 2 {1} { } 3 {2, 1}
( ) A ( ) B ( ) C ( ) D
LexDFS { ,1} 2 {2, 1} 3 1 4 {3} {2, 1} 5 {4, 2, 1} 3 1 4 {3} {2, 1} { , 2, 1} 4 ( ) E {0} {0} {0} 2 {1} 2 {1}
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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
MCS
(The following algorithm cut from Reference [Berry et al., 2010]) 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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
An exmaple for MCS
2 1 1 1 3 2 1
( ) A ( ) B ( ) C ( ) D
MCS ( ) E 1 1 1 2 2 1 2 2 1 3 1 2 2 1 4 3 1 2 2 1 4 5 2 2 2 2
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
MNS
(The following algorithm cut from Reference [Corneil et al., 2008]) D.G. Corneil, R.M. krueger, A unified view of graph searching, SIAM J. Discrete
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms
LBFS LDFS MCS MNS notes on algorithms
Properties
vertcies End-vertex Problems
Notes on algorithms
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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Some basic definitions
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
A chordal graph is a graph with no chordless cycle C of length greater than 3, i.e., there is an edge connecting nonconsecu- tive 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. A vertex v is admissible if there are no unrelated vertices with
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Applications of end-verties of LBFS
(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 dom- inating 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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Basic properties of end-vertices
(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 AT- free 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.
D.G. Corneil, R.M. krueger, A unified view of graph searching, SIAM J. Discrete
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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Basic definitions about Moplex
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
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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Illustrations for definitions, such as, moplex
both a clique and a module and A is inclusion-maximal for both properties.
a minimal separator.
S1 S2 V1 V9 V8 V7 V6 V5 V4 V3 V2
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}.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Notes on moplex
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 simpli- cial, 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])
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Berry et al. results on moplex
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.
Algorithms, Algorithms, 3 (2010) 100–124.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Illustration for Berry et al. rusults
(The following figure cut and adapted from Reference [Berry et al., 2010])
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}.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Xu et al. results on moplex
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 mini- mal 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
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Counterexamples
1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 (A) (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}.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies
Some basic definitions Applications
end-vertices
Properties of end-vertices Basic properties of end-vertices Moplex property of end-vertices Moplex property of End-vertices
Moplex properties of end-vertices
End-vertex
Moplex property of end-vertices of MCS
MCS on a chordal graph ends on a vertex which belongs to a moplex. On a non-chordal graph?? WRONG
1 2 3 4 5 6
The end-vertex 6 is not a vertex of a moplex.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies End-vertex Problems
Characterizations
end-vertices Complexity
Problems
Characterizations of end-vertices
(Corneil et al., 2009) A vertex of an interval graph is an end- vertex 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
(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 min- imal 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.
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies End-vertex Problems
Characterizations
end-vertices Complexity
Problems
Descriptions for End-vertex Problems
(The following problems cut from Reference [Charbit et al., 2014])
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies End-vertex Problems
Characterizations
end-vertices Complexity
Problems
Complexity of end-vertex problem
(Charbit et al., 2014) The end-vertex and the beginning-end- vertex 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 poly- nomial 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.
Algorithms, Algorithms, 3 (2010) 100–124.
problem, http://www.liafa.univ-paris-diderot.fr/ mamcarz/papers/endvertices.pdf
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies End-vertex Problems
Characterizations
end-vertices Complexity
Problems
A summary of Carbit et al. results on complexity
(The following table cut from Reference [Charbit et al., 2014])
problem, http://www.liafa.univ-paris-diderot.fr/ mamcarz/papers/endvertices.pdf
Report in Shanghai Shou-Jun Xu (MÅ ) Graph searching algorithms Properties
vertcies End-vertex Problems
Characterizations
end-vertices Complexity
Problems