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Graph isomorphism is in the low hierarchy

Uwe Schöning
1988 Journal of computer and system sciences (Print)  
It is shown that the graph isomorphism problem is located in level L$ of the low hierarchy in NP.  ...  This implies that this problem is not NP-complete (even under weaker forms of polynomial reducibilities, like y-reducibility) unless the polynomial-time hierarchy collapses to some finite level.  ...  The graph isomorphism problem is given as the set GRAPH ISOMORPHISM = ((G,, G,) 1 G, and G, are isomorphic} which is easily seen to belong to the class NP, but is currently not known to be NP-complete  ... 
doi:10.1016/0022-0000(88)90010-4 fatcat:4l6j2onzijbslnow4v6g34m3mu

Isomorphic Subgraphs [chapter]

Sabine Bachl
1999 Lecture Notes in Computer Science  
We show that the Isomorphic Subgraphs problem is NP-hard for connected outerplanar graphs, and 2-connected planar graphs and is solvable in linear time when restricted to trees.  ...  There are many NP-complete or open problems related to our problem, like Graph Isomorphism, Graph Automorphism or Largest Common Subgraph.  ...  Brandenburg for his various and detailed comments and suggestions, particularly on the proof of Isomorphic Subgraphs of general graphs.  ... 
doi:10.1007/3-540-46648-7_30 fatcat:wiec3yfyk5agpeweik3zz3ybby

Page 662 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
ISBN 0-8176-3680-3 The graph isomorphism problem has received much recent atten- tion. It is known that if P does not equal NP then there must exist problems that are neither NP-complete nor in P.  ...  The authors derive from the interactive proof theory 05 COMBINATORICS 662 the fact that if the graph isomorphism problem is NP-complete, then the polynomial-time hierarchy collapses.  ... 

Subgraph isomorphism in graph classes

Shuji Kijima, Yota Otachi, Toshiki Saitoh, Takeaki Uno
2012 Discrete Mathematics  
We investigate the computational complexity of the following restricted variant of Subgraph Isomorphism: given a pair of connected graphs G =  ...  Part of this research is supported by the Funding Program for World-Leading Innovative R & D on Science and Technology, Japan, and Grants-in-Aid for Scientific Research from Ministry of Education, Science  ...  Spanning Subgraph Isomorphism is NP-complete even for bipartite graphs and for chordal graphs, since Hamiltonian Path on these classes is NP-complete [10, 1] .  ... 
doi:10.1016/j.disc.2012.07.010 fatcat:cxupp725szgvbj5htfpuc6vuba

Square-Root Finding Problem In Graphs, A Complete Dichotomy Theorem [article]

Babak Farzad, Majid Karimi
2012 arXiv   pre-print
Determining if a given graph G is the square of some graph is not easy in general. Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph.  ...  We introduce a family of graphs with exponentially many non-isomorphic square roots, and as the main result of this paper we prove that the square root finding problem is NP-complete for square roots of  ...  r for r = 2k + 1 is NP-complete.  ... 
arXiv:1210.7684v1 fatcat:75g3etssonfa5e3gpqgl4itqna

Induced Subtrees in Interval Graphs [chapter]

Pinar Heggernes, Pim van 't Hof, Martin Milanič
2013 Lecture Notes in Computer Science  
This problem is known to be NP-complete on bipartite graphs, but it can be solved in polynomial time when G is a forest.  ...  In contrast to this positive result, we show that the closely related Subtree Isomorphism problem is NP-complete even when G is restricted to the class of proper interval graphs, a well-known subclass  ...  Note that the problem is NP-complete on perfect graphs, a superclass of chordal graphs, due to the aforementioned NP-completeness of Induced Subtree Isomorphism on bipartite graphs [7] .  ... 
doi:10.1007/978-3-642-45278-9_20 fatcat:yddoddstejeohk5nb4v3zc5uvu

Page 3786 of Mathematical Reviews Vol. , Issue 84i [page]

1984 Mathematical Reviews  
It is shown that for many very simple undirected graphs G this question is NP-complete (this was previously known for the graphs K,, only).  ...  Inform. and Control 51 (1981), no. 2, 128-145. Authors’ summary: “A graph H is called an interpretation of a graph G if a morphic image of H is (isomorphic to) a subgraph of G.  ... 

The complexity of comparing reaction systems

M. Ettinger
2002 Bioinformatics  
We show that comparing the stoichiometric structure of two reactions systems is equivalent to the graph isomorphism problem.  ...  The analogous problem of searching for a subsystem of a reaction system is NP-complete. We also discuss heuristic issues in implementations for practical comparison of stoichiometric matrices.  ...  Matrix isomorphism is reducible to graph isomorphism and therefore is GI-complete. We have mentioned that submatrix isomorphism is NP-complete.  ... 
doi:10.1093/bioinformatics/18.3.465 pmid:11934746 fatcat:tatwnrca6ndhjgj2hvfvihpqqa

The Subgraph Bisimulation Problem

A. Dovier, C. Piazza
2003 IEEE Transactions on Knowledge and Data Engineering  
We study the complexity of the Subgraph Bisimulation Problem, which stands to Graph Bisimulation as Subgraph Isomorphism stands to Graph Isomorphism and we prove its NP-Completeness.  ...  Our analysis is motivated by its applications to Semistructured Databases.  ...  Each Hamilton path on a 5-nodes graph is isomorphic (hence, bisimilar) to the 5-chain C 5 . Theorem 1: The Subgraph Bisimulation problem is NP-complete.  ... 
doi:10.1109/tkde.2003.1209024 fatcat:vmw6gqlyizdq7jhfu64pv3kxfi

The complexity of Boolean matrix root computation

Martin Kutz
2004 Theoretical Computer Science  
related to subdivision digraphs, root finding is of the same complexity as the graph-isomorphism problem.  ...  Interpreting Boolean matrices as directed graphs, we further reveal a connection between Boolean matrix roots and graph isomorphism, which leads to a proof that for a certain subclass of Boolean matrices  ...  for graphs is NP-complete [5] .  ... 
doi:10.1016/j.tcs.2004.02.041 fatcat:t56g4mzfezhepofgd3ix7m3pau

The Complexity of Boolean Matrix Root Computation [chapter]

Martin Kutz
2003 Lecture Notes in Computer Science  
related to subdivision digraphs, root finding is of the same complexity as the graph-isomorphism problem.  ...  Interpreting Boolean matrices as directed graphs, we further reveal a connection between Boolean matrix roots and graph isomorphism, which leads to a proof that for a certain subclass of Boolean matrices  ...  for graphs is NP-complete [5] .  ... 
doi:10.1007/3-540-45071-8_23 fatcat:qhcl7kreuvg3vkotbbskjnhfpy

On the complexity of deciding whether the distinguishing chromatic number of a graph is at most two

Elaine M. Eschen, Chính T. Hoàng, R. Sritharan, Lorna Stewart
2011 Discrete Mathematics  
We consider the problem when k = 2. In regards to the issue of solvability in polynomial time, we show that the problem is at least as hard as graph automorphism, but no harder than graph isomorphism.  ...  In an article Cheng (2009) [3] published recently in this journal, it was shown that when k ≥ 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard.  ...  Graph automorphism and graph isomorphism Consider the following decision problems each of which is known to be in NP, but neither of which is known to be in P or NP-complete.  ... 
doi:10.1016/j.disc.2010.12.013 fatcat:5e2rkmi4v5axniblk5dyzehwsy

A note on contracting claw-free graphs

Jiří Fiala, Marcin Kamiński, Daniël Paulusma
2013 Discrete Mathematics & Theoretical Computer Science  
We show that these problems stay NP-complete even when the host and target belong to the class of line graphs, which form a subclass of the class of claw-free graphs, i.e., graphs with no induced 4-vertex  ...  A natural question is to study the computational complexity of these problems if the target graph is assumed to be fixed.  ...  Because DISSOLUTION is NP-complete, we then find that this problem stays NP-complete even when both the host and target graph are line graphs. We now consider the GRAPH ISOMORPHISM problem.  ... 
doi:10.46298/dmtcs.605 fatcat:hrmcp2vu7jehnm432miqaeucam

The complexity of equivalence and isomorphism of systems of equations over finite groups

Gustav Nordh
2005 Theoretical Computer Science  
If the group is Abelian, then the isomorphism problem is GRAPH ISOMORPHISM-hard.  ...  We show that the equivalence problem is in P if the group is Abelian, and coNP-complete if the group is non-Abelian.  ...  problem is GRAPH ISOMORPHISM-complete.  ... 
doi:10.1016/j.tcs.2005.07.018 fatcat:63za63ve5rhsrhidxnz7b4xwbq

Isomorphic Implication [article]

Michael Bauland, Edith Hemaspaandra
2005 arXiv   pre-print
We prove that, depending on the set of constraints, this problem is in P, NP-complete, or NP-hard, coNP-hard, and in parallel access to NP.  ...  We study the isomorphic implication problem for Boolean constraints. We show that this is a natural analog of the subgraph isomorphism problem.  ...  The subgraph isomorphism problem for graphs without isolated vertices is NP-complete.  ... 
arXiv:cs/0412062v2 fatcat:zf326f3lj5cjtl6h5zwnwu6t7u
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