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13
Directed planar reachability is in unambiguous logspace
 In Proceedings of IEEE Conference on Computational Complexity CCC
, 2007
"... We show that the stconnectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1. ..."
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Cited by 22 (6 self)
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We show that the stconnectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1.
3connected planar graph isomorphism is in logspace
, 2008
"... We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. ..."
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Cited by 13 (3 self)
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We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13].
Reachability in K3,3free graphs and K5free graphs is in unambiguous logspace
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2009
"... We show that the reachability problem for directed graphs that are either K3,3free or K5free is in unambiguous logspace, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL. Our algorithm de ..."
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Cited by 12 (2 self)
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We show that the reachability problem for directed graphs that are either K3,3free or K5free is in unambiguous logspace, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL. Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The nonplanar components are replaced by planar components that maintain the reachabilty properties. For K5free graphs we also need a decomposition into fourconnected components. A careful analysis finally gives a polynomial size planar graph which can be computed in logspace. We show the same upper bound for computing distances in K3,3free and K5free directed graphs and for computing longest paths in K3,3free and K5free directed acyclic graphs.
A Logspace Algorithm for Partial 2Tree canonization
, 2008
"... We show that partial 2tree canonization, and hence isomorphism testing for partial 2trees, is in deterministic logspace. Our algorithm involves two steps: (a) We exploit the “tree of cycles ” property of biconnected partial 2trees to canonize them in logspace. (b) We analyze Lindell’s tree cano ..."
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Cited by 7 (1 self)
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We show that partial 2tree canonization, and hence isomorphism testing for partial 2trees, is in deterministic logspace. Our algorithm involves two steps: (a) We exploit the “tree of cycles ” property of biconnected partial 2trees to canonize them in logspace. (b) We analyze Lindell’s tree canonization algorithm and show that canonizing general partial 2trees is also in logspace, using the algorithm to canonize biconnected partial 2trees.
The Complexity of Planar Graph Isomorphism
, 1997
"... The Graph Isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. In terms of complexity classes however, the exact complexity of the problem has been established only very recently. It was proved in [6] that planar graph isomorphism can be co ..."
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Cited by 3 (0 self)
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The Graph Isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. In terms of complexity classes however, the exact complexity of the problem has been established only very recently. It was proved in [6] that planar graph isomorphism can be computed within logarithmic space. Since there is a matching hardness result [12], this shows that the problem is complete for L. Although this could be considered as a result in algorithmics its proof relies on several important new developments in the area of logarithmic space complexity classes and reflects the close connections between algorithms and complexity theory. In this column we give an overview of this result mentioning the developments that led to it.
On the Power of Unambiguity in Logspace
, 2010
"... We report progress on the NL vs UL problem. We show unconditionally that the complexity class ReachFewL ⊆ UL. This improves on the earlier known upper bound ReachFewL ⊆ FewL. We investigate the complexity of minuniqueness a central notion in studying the NL vs UL problem. – We show that minuniq ..."
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Cited by 3 (2 self)
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We report progress on the NL vs UL problem. We show unconditionally that the complexity class ReachFewL ⊆ UL. This improves on the earlier known upper bound ReachFewL ⊆ FewL. We investigate the complexity of minuniqueness a central notion in studying the NL vs UL problem. – We show that minuniqueness is necessary and sufficient for showing NL = UL. – We revisit the class OptL[log n] and show that ShortestPathLength computing the length of the shortest path in a DAG, is complete for OptL[log n]. – We introduce UOptL[log n], an unambiguous version of OptL[log n], and show that (a) NL = UL if and only if OptL[log n] = UOptL[log n], (b) LogFew ≤ UOptL[log n] ≤ SPL. We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages which is logspace equivalent to the reachability problem in planar graphs and hence is in UL.
A logspace algorithm for canonization of planar graphs
, 2008
"... Planar graph canonization is known to be hard for L this directly follows from Lhardness of treecanonization [Lin92]. We give a logspace algorithm for planar graph canonization. This gives completeness for logspace under AC 0 manyone reductions and improves the previously known upper bound of A ..."
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Cited by 2 (1 self)
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Planar graph canonization is known to be hard for L this directly follows from Lhardness of treecanonization [Lin92]. We give a logspace algorithm for planar graph canonization. This gives completeness for logspace under AC 0 manyone reductions and improves the previously known upper bound of AC 1 [MR91]. A planar graph can be decomposed into biconnected components. We give a logspace procedure for the decomposition of a biconnected planar graph into a triconnected component tree. The canonization process is based on these decomposition steps. 1
Longest Paths in Planar DAGs in Unambiguous Logspace
, 2009
"... Reachability and distance computation are known to be NLcomplete in general graphs, but within UL ∩ coUL if the graphs are planar. However, finding longest paths is known to be NPcomplete, even for planar graphs. We show that with the combination of planarity and acyclicity, finding the length of ..."
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Cited by 1 (0 self)
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Reachability and distance computation are known to be NLcomplete in general graphs, but within UL ∩ coUL if the graphs are planar. However, finding longest paths is known to be NPcomplete, even for planar graphs. We show that with the combination of planarity and acyclicity, finding the length of the longest path (and also enumerating one such path) is also in UL ∩ coUL. The result extends to toroidal DAGs as well. We also address the question of when reachability, distance, and longest path are indeed equivalent on DAGs, and give partial bounds. When the number of distinct paths is bounded by a polynomial, counting the number of paths is known to be in NL. We show that for planar DAGs with this promise, counting can be done by a UAuxPDA in polynomial time. The UAuxPDA(poly) bound also holds if we want to compute the number of longest paths, or shortest paths, and this number is bounded by a polynomial (irrespective of the total number of paths). Along the way, we show that counting in general DAGs is possible in LogDCFL provided the number of paths is bounded by a polynomial and the target node is the only sink.
Reachability in K3,3free and K5free Graphs is in Unambiguous Logspace
, 2014
"... We show that the reachability problem for directed graphs that are either K3,3free or K5free is in unambiguous logspace, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL. Our algorithm d ..."
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Cited by 1 (0 self)
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We show that the reachability problem for directed graphs that are either K3,3free or K5free is in unambiguous logspace, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL. Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The nonplanar components are replaced by planar components that maintain the reachability properties. For K5free graphs we also need a decomposition into 4connected components. Thereby we provide a logspace reduction to the planar reachability problem. We show the same upper bound for computing distances in K3,3free and K5free directed graphs and for computing longest paths in K3,3free and K5free directed acyclic graphs.
Hardness results for isomorphism and automorphism of bounded valence graphs
 In Theory and Practice of Computer Science (SOFSEM), volume 2  Stutend Research Forum
, 2008
"... Abstract. In a bounded valence graph every vertex has O(1) neighbours. Testing isomorphism of bounded valence graphs is known to be in P [15], something that is not clear to hold for graph isomorphism in general. We show that testing isomorphism for undirected, directed and colored graphs of valenc ..."
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Abstract. In a bounded valence graph every vertex has O(1) neighbours. Testing isomorphism of bounded valence graphs is known to be in P [15], something that is not clear to hold for graph isomorphism in general. We show that testing isomorphism for undirected, directed and colored graphs of valence 2 is logspace complete. We also prove the following: If a special version of bounded valence GI is hard for ModkL then it is also hard for #L. All results are proved with respect to DLOGTIME uniform AC0 manyone reductions. 1