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O(m logn) Split Decomposition of Strongly Connected Graphs
[chapter]

2009
*
Lecture Notes in Computer Science
*

In the early 1980's, Cunningham described a unique

doi:10.1007/978-3-642-02029-2_16
fatcat:chdvtrj7uzg6xmsdfwpef4mmfe
*decomposition**of*a*strongly*-*connected**graph*. ... We give an*O*(*m*log n) bound. ... This leaves open the possibility*of*improving on the previous best bound*of**O*(n 3 ) for finding the*decomposition**of**strongly*-*connected*digraphs. In this paper, we give an*O*(*m*log n) bound. ...##
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Quadtree Decomposition, Steiner Triangulation, and Ray Shooting
[chapter]

1998
*
Lecture Notes in Computer Science
*

For a polygon

doi:10.1007/3-540-49381-6_39
fatcat:facad6j7t5carbw77tvedtsrpq
*of*n vertices, a truncated*decomposition*can be computed in*O*(nlog n) time which yields a Steiner triangulation*of*the interior*of*the polygon that has*O*(n log n) size and approximates the ... We present a new quadtree-based*decomposition**of*a polygon possibly with holes. ... The total complexity*of*is*O*(*m*) as any spanning tree edge bounding is incident to a vertex*of*. This implies that the clipping takes*O*(*m*) time. ...##
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Improved Deterministic Algorithms for Decremental Transitive Closure and Strongly Connected Components
[chapter]

2011
*
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
*

In addition to that, a few decremental algorithms for maintaining

doi:10.1137/1.9781611973082.111
dblp:conf/soda/Lacki11
fatcat:sn5vvohn2rh3zd2gzprswcpmz4
*strongly**connected*components are shown, whose time complexity is*O*(n 1.5 ) for planar*graphs*,*O*(n log n) for*graphs*with bounded treewidth ... This paper presents a new deterministic algorithm for decremental maintenance*of*the transitive closure in a directed*graph*. ... Corollary 2 . 1 . 21 There exists a deterministic algorithm for decremental maintenance*of**strongly**connected*components, which runs in*O*(mn) total time and answers queries in*O*(1) time, using*O*(n +*m*) ...##
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Page 1995 of Mathematical Reviews Vol. , Issue 84e
[page]

1984
*
Mathematical Reviews
*

Authors’ summary: “In a recent paper an algorithm for generating maximal minimally

*strongly**connected*subgraphs or cliques in undirected*graphs*was proposed by Das et al. (Comput. Electr. ... Here n is the number*of*vertices and*m*is the number*of*edges*of*a*graph*. This algorithm can be used in many problems. ...##
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Page 8210 of Mathematical Reviews Vol. , Issue 99m
[page]

1999
*
Mathematical Reviews
*

We show that every cograph has a perfect path cover, and we exploit this to obtain an

*O*(*m*+n*logn*)-time algorithm for finding, for any arbitrary k, a k-blanket or a k-core*of*an arbitrary cograph on n ... Both problems are NP-complete for various re- stricted classes*of**graphs*including chordal bipartite*graphs*and*split**graphs*. ...##
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Modular decomposition and transitive orientation

1999
*
Discrete Mathematics
*

We give

doi:10.1016/s0012-365x(98)00319-7
fatcat:of5qq4jgdzgobbs5ehniofn6qe
*O*(n +*m*) algorithms for modular*decomposition*and transitive orientation, where n and*m*are the number*of*vertices and edges*of*the*graph*. ... There is a canonical linear-space representation for the modules*of*a*graph*, called the modular*decomposition*. ... Prime*graphs*are*connected*, so n =*O*(*m*) and the bound reduces to*O*(*m*log n). ...##
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Efficient parallel recognition algorithms of cographs and distance hereditary graphs

1995
*
Discrete Applied Mathematics
*

As a consequence distance hereditary

doi:10.1016/0166-218x(93)e0138-o
fatcat:evlufc7abndi5fh4vlv2owxkgq
*graphs*can be recognized by the same processor and time bound. ... This result extends the result*of*Adhar and Peng (1990). Moreover, we get a better time processor product than the algorithm*of*Lin and Olariu (199 I ). ... Step 4 can be executed in*O*(log' n) time and*O*(*M*+*m*) processors by Theorem 1. In step 5 we first compute the set*of*Sj (*O*(*logn*) time and*O*(n +*m*) processors). ...##
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A framework for solving VLSI graph layout problems

1984
*
Journal of computer and system sciences (Print)
*

It is also shown how good

doi:10.1016/0022-0000(84)90071-0
fatcat:5nbwje5u6zd43exgklsvakve7q
*graph*partitioning heuristics may be used to develop a provably good layout strategy. ... The framework is also effectively used to design regular and configurable layouts, to assemble large networks*of*processors using restructurable chips, and to configure networks around faulty processors ... Since any two sets*of*rl and r2 nodes are linked by a path*of*length*O*(log(*m*/r,) + log(*m*/r,)) in an*m*-node expander*graph*, this means that there is a path*of*length*O*(log n)*connecting*the left outer strip ...##
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On subdirectly irreducible automata

1981
*
RAIRO Informatique théorique
*

The number

doi:10.1051/ita/1981150100231
fatcat:num4gsc7xvcc7dwh6znf3wsgxe
*of*edges in the new*strongly**connected*component is at most*m**m*Since*m*^2, the new*strongly**connected*component having s-Y, s t vertices, has at most 2s -2 edges. ... In this case c(G i ) = c(G i+i ) and there is no new edge inside the*strongly**connected*components*of*G i + 1 ; b) the*graph*C (G f ) with the added edge contains a cycle consisting*of**strongly**connected*...##
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General spanning trees and reachability query evaluation

2009
*
Proceedings of the 2009 C3S2E conference on - C3S2E '09
*

n + e + t 3 )

doi:10.1145/1557626.1557665
dblp:conf/c3s2e/Chen09
fatcat:acape3om7bfkjn6azjqwzowusy
*O*(n 4 )*O*(n 2.376 ) 0 ours*O*(logb)*O*(be)*O*(bn)*O*(k 1 n)*O*(n + t 2 )*O*(ne 1/2*logn*)*O*(n 2 )*O*(bn) (Jagadish) dual-I*O*(1)*O*(n + e + t 3 )*O*(n + t 2 )*graph*-*decomposition**O*(logk 2 )*O*(k 2 e) ... For a cyclic*graph*(a*graph*containing cycles), we can find all the*strongly**connected*components (SCC) in linear time [24] and then collapse each*of*them into a representative node. ... However, a cannot be*connected*to d. The final result is a set*of*six chains as shown in Fig. 14(e) . But the*graph*can be decomposed into a set*of*five chains as shown in Fig. 14(f) . ...##
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Preemptive Ensemble Motion Planning on a Tree

1992
*
SIAM journal on computing (Print)
*

*m*. ... The first algorithm runs in

*O*(k +qn) time, where n is the number

*of*vertices in the tree, k 1s the number

*of*objects to be moved, and q 5 min {k, n} is the number

*of*nontrivial

*connected*components in ... Our first algorithm runs in

*O*(k + qn) time, where q:$ min{k1n} is the number

*of*nontrivial

*strongly*

*connected*components in a related directed

*graph*. ...

##
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An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

2017
*
Theory and Applications of Graphs
*

Li and Sun: Rainbow

doi:10.20429/tag.2017.000103
fatcat:76awzxcw6ra3ldnibx5xtvtmmu
*connections**of**graphs*-a dynamic survey Li and Sun: Rainbow*connections**of**graphs*-a dynamic survey Published by Digital Commons@Georgia Southern, 2017, Last updated 2017 σ 2 . ... Li and Liu got a best possible upper bound in [95, 97] for 2-*connected**graphs*: Let G be a 2-*connected**graph**of*order n (n ≥ 3), then rc(G) ≤ ⌈ n 2 ⌉ and the upper bound is tight for n ≥ 4. ... Acknowledgement: The authors would like to thank the referees and the editor for their helpful suggestions and comments which helped to polish the presentation*of*the paper. ...##
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The geometry of graphs and some of its algorithmic applications

1995
*
Combinatorica
*

In this paper we explore some implications

doi:10.1007/bf01200757
fatcat:lhkhob7abvhc7bhcfzyci7uqoy
*of*viewing*graphs*as geometric objects. This approach offers a new perspective on a number*of**graph*-theoretic and algorithmic problems. ... There are several ways to model*graphs*geometrically and our main concern here is with geometric representations that respect the metric*of*the (possibly weighted)*graph*. ... Remark 6.2: Combined with Theorem 3.5 we obtain a*decomposition*to*O*(1ogn) blocks*of*diameter*O*(10g3 n), a result slightly inferior to the optimal (*logn*,*logn*) from [34]. ...##
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A spectral method for community detection in moderately-sparse degree-corrected stochastic block models
[article]

2017
*
arXiv
*
pre-print

We show that this algorithm consistently recovers the block-membership

arXiv:1506.08621v3
fatcat:6oh3gdhrmvalbp4yb74z4wqf6e
*of*all but a vanishing fraction*of*nodes, in the regime where the lowest degree is*of*order log(n) or higher. ... In particular, it does not need to know the number*of*communities. ... Further, we point out a natural*connection*between H and a random walk on the observed*graph*. ...##
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Structure and Hardness in P (Dagstuhl Seminar 16451)

2017
*
Dagstuhl Reports
*

The main goal

doi:10.4230/dagrep.6.11.1
dblp:journals/dagstuhl-reports/LewensteinPW16
fatcat:hha7ev23anezhi37biylsyfxie
*of*the seminar was to bring together researchers from several disciplines and*connect*those who work on proving conditional lower bounds with those who or may benefit from it. ... This resulted in an extensive list*of*open problems which is also provided. ... We study the complexity*of*computing the diameter and other distance measures in an unweighted, undirected*graph*. ...
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