Report in Shanghai Shou-Jun Xu ( M Å � ) End-vertices of graph searching algorithms Graph searching algorithms Shou-Jun Xu ( M Å � ) Properties of end- vertcies E-mail:shjxu@lzu.edu.cn End-vertex Lanzhou University Problems @ Shanghai Jiao Tong University, China Jun 6, 2014
Outlines Report in Shanghai Shou-Jun Xu ( M Å � ) Graph Graph searching algorithms 1 searching algorithms Properties of end- vertcies Properties of end-vertcies 2 End-vertex Problems End-vertex Problems 3
Graph searching algorithms Report in Shanghai • General graph searching algorithm: employing some mechanism Shou-Jun Xu ( M Å for systematically visiting all vertices and edges in the given graph. � ) After choosing an initial vertex, a search of a connected graph visits Graph each of the vertices and edges of the graph such that a new vertex searching algorithms is visited only if it is adjacent to some previously visited vertex. LBFS LDFS • Several well-known graph searching algorithms MCS MNS notes on BFS (Breadth-first search); algorithms Properties DFS (Depth-first search); of end- vertcies LBFS (Lexicographic BFS, Rose et al., 1976); End-vertex Problems LDFS (Lexicographic DFS, Corneil et al., 2008); MCS (Maximum cardinality searching, Tarjan et al., 1984); MNS (Maximal neighborhood searching, Corneil et al., 2008).
LBFS Report in Shanghai (The following algorithm cut from Reference [Corneil et al., 2008]) Shou-Jun Xu ( M Å � ) Graph searching algorithms LBFS LDFS MCS MNS notes on algorithms Properties of end- vertcies 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 Report in Shanghai {5-1} {4} {4, 5-3 } Shou-Jun Xu ( M Å � ) 1 1 3 1 {5-1} {6} {4, 5-2 } {6} { 5-2 } {6} { 3} {4, 3} Graph searching algorithms {5-1} LBFS {4} 2 {4} ( ) 2 A ( ) B ( C ) LDFS MCS {4, 2} 4 {4, 2} 4 MNS notes on algorithms Properties 3 1 3 1 of end- { 3, 5-4 } {6} {4, 3} {6} 5 { 3, 1} {4, 3} vertcies End-vertex Problems {4} ( E ) 2 ( D ) 2 {4} LexBFS • Note that at any step, if we denote l ( u ) = ( a 1 , a 2 , · · · , a k ) , then a i 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 a 1 first, a 2 second, and so on.
LDFS Report in Shanghai (The following algorithm cut from Reference [Corneil et al., 2008]) Shou-Jun Xu ( M Å � ) Graph searching algorithms LBFS LDFS MCS MNS notes on algorithms Properties of end- vertcies 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 Report in Shanghai {1} { ,1} 2 3 {2, 1} Shou-Jun Xu ( M Å � ) 1 1 1 {1} {0} 2 {1} {0} { } 3 {0} 2 {1} Graph searching algorithms {1} { ,1} 2 LBFS {2, 1} ( A ) ( B ) 3 {2, 1} ( C ) LDFS 3 {2, 1} MCS MNS notes on 1 algorithms 1 2 {1} 4 {3} {0} Properties 2 {1} 4 {3} {0} of end- vertcies 5 {4, 2, 1} LexDFS ( E ) { , 2, 1} 4 ( D ) End-vertex Problems • Note that at any step, if we denote l ( u ) = ( a 1 , a 2 , · · · , a k ) , then a i 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 a k first, a k − 1 second, and so on.
MCS Report in (The following algorithm cut from Reference [Berry et al., 2010]) Shanghai Shou-Jun Xu ( M Å � ) Graph searching algorithms LBFS LDFS MCS MNS notes on algorithms Properties of end- vertcies 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 Report in Shanghai Shou-Jun Xu ( M Å � ) 1 2 3 2 Graph searching 1 1 algorithms 1 1 1 2 0 0 2 1 1 LBFS 0 LDFS MCS MNS 1 2 2 ( A ) notes on ( ) B 3 2 ( ) C algorithms 3 2 Properties of end- 1 vertcies 1 2 1 0 2 2 1 0 2 5 End-vertex Problems 4 ( ) 2 D 2 ( ) MCS 4 E
MNS Report in Shanghai (The following algorithm cut from Reference [Corneil et al., 2008]) Shou-Jun Xu ( M Å � ) Graph searching algorithms LBFS LDFS MCS MNS notes on algorithms Properties of end- vertcies 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 Report in Shanghai Shou-Jun Xu ( M Å � ) Graph searching • Let G be a connected graph with vertex set V of size n . We algorithms LBFS define α = ( α (1) , α (2) , · · · , α ( n )) to be a linear ordering of V , LDFS MCS which will typically be the left-to-right order in which we number MNS notes on the vertices in a search, the smaller the number is labeled, the algorithms Properties earlier the vertex is visited. of end- vertcies End-vertex Problems
Some basic definitions Report in Let G be a connected graph. A vertex v in G is called an Shanghai end-vertex of some graph searching algorithm (such as LBFS, Shou-Jun Xu ( M Å LDFS), if v is visited last by some execution of the algorithm � ) on G . Graph A chordal graph is a graph with no chordless cycle C of length searching algorithms greater than 3, i.e., there is an edge connecting nonconsecu- Properties tive vertices on any cycle of length greater than 3. of end- vertcies A clique is a set of pairwise adjacent vertices. Some basic definitions A vertex is simplicial if its neighborhood is a clique. Applications of A path P misses vertex v (or v misses P ) if P ∩ N ( v ) = ∅ end-vertices of LBFS Properties of (i.e., no vertex of P is adjacent to v ). end-vertices Basic Two vertices x, y are unrelated with respect to vertex v if properties of end-vertices Moplex there are paths P between x and v and Q between y and v property of end-vertices such that P misses y and Q misses x . Moplex property of End-vertices An independent triple of vertices x, y, z is an Asteroidal triple of LBFS Moplex (AT), if between every pair of vertices, there is a path that properties of end-vertices of LDFS misses the third. End-vertex A vertex v is admissible if there are no unrelated vertices with
Applications of end-verties of LBFS Report in (Dragan et al., 1997) The eccentricity ecc ( v ) of an end-vertex Shanghai v is equal to the graph’s diameter diam ( G ) if G is interval. Shou-Jun Xu ( M Å ecc ( v ) � diam ( G ) − 1 if G is HHD-free (Dragan, 1999), � ) chordal (Dragan et al. 1997) or AT-free (Corneil, 2001). Graph searching ecc ( v ) � diam ( G ) − 2 if G is HH-free (Dragan, 1999). algorithms For these restricted families of graphs, there are an easy, linear Properties of end- time algorithm that closely approximates the diameter of the vertcies Some basic graph. definitions Applications of (Corneil et al, 1999) End-vertices can be used to find a dom- end-vertices of LBFS inating pair in connected AT-free graph G . Properties of end-vertices Basic properties of F.F. Dragan, F. Nicolai, A. Brandstadt, LDFS-orderings and powers of graphs, in: end-vertices Moplex LNCS 1197 (1997) 166–180. property of end-vertices F.F. Dragan, Almost diameter of a house-hole-free graph in linear time via LBFS, Moplex property of End-vertices Discrete Appl. Math. 95 (1999) 223-239. of LBFS Moplex properties of D.G. Corneil, F.F. Dragan, M. Habib, C. Paul, Diameter determination on restricted end-vertices of LDFS graph families, Discrete Appl. Math. 113 (2001) 143-166. End-vertex
Basic properties of end-vertices Report in (Rose et al., 1976) The end-vertex of an LBFS of a chordal Shanghai graph is simplicial. Shou-Jun Xu ( M Å (Tarjan, Yannakakis, 1984) The end-vertex of an MCS of a � ) chordal graph is simplicial. Graph (Corneil et al., 2008) The end vertex of an MNS (say an searching algorithms LDFS) of a chordal graph is simplicial. Properties (Corneil et al., 1999) The end-vertex of an LBFS of an AT- of end- vertcies free graph is admissible. Some basic definitions Applications D.J. Rose, R.E. Tarjan, G.S. Lueker, Algorithmic aspects of vertex elimination on of end-vertices graphs, SIAM J. Comput. 5 (1976) 266–283. of LBFS Properties of end-vertices R.E. Tarjan, M. Yannakakis, Simple linear-time algorithms to test chodality of graphs, Basic properties of test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. end-vertices Moplex property of Comput. 13 (1984) 566–579. end-vertices Moplex D.G. Corneil, R.M. krueger, A unified view of graph searching, SIAM J. Discrete property of End-vertices of LBFS Math. 22 (2008) 1259–1276. Moplex properties of D.G. Corneil, S. Olariu, L. Stewart, Linear time algorithms for dominating pairs in end-vertices of LDFS asteroidal triple-free graphs, SIAM J. Comput 28 (1999) 1284–1297. End-vertex
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