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A reductive technique for enumerating non-isomorphic planar maps

Valery A. Liskovets
1996 Discrete Mathematics  
A general method for the counting of unrooted planar maps is proposed. It reduces the problem to the counting of rooted maps of several classes of three kinds: planar, projective and 'circular'.  ...  A general formula of enumerating under orthogonal actions of a group is also derived. V.A. Liskovets / Discrete Mathematics 156 (1996) 197 217 2.3.  ...  This general theorem reduces the enumeration of non-isomorphic planar maps to that of planar, projective and circular rooted maps of several kinds.  ... 
doi:10.1016/0012-365x(94)00347-l fatcat:xkmkur3mpnadlpsonepgwcutk4

Efficient Generation of Stable Planar Cages for Chemistry [article]

Dominique Barth, Olivier David, Franck Quessette, Vincent Reinhard, Yann Strozecki, Sandrine Vial
2015 arXiv   pre-print
In this paper we describe an algorithm which generates all colored planar maps with a good minimum sparsity from simple motifs and rules to connect them.  ...  We first build for each letter aA a list L a of all non isomorphic motifs whose first edge is incident to a vertex of labelā.  ...  The drawback is that not any planar graph has an Hamiltonian path, therefore we could miss some planar maps in our enumeration.  ... 
arXiv:1503.06610v1 fatcat:ywb4wcysrbdnbhuzo7t3ihyyf4

Efficient Generation of Stable Planar Cages for Chemistry [chapter]

Dominique Barth, Olivier David, Franck Quessette, Vincent Reinhard, Yann Strozecki, Sandrine Vial
2015 Lecture Notes in Computer Science  
In this paper we describe an algorithm which generates all colored planar maps with a good minimum sparsity from simple motifs and rules to connect them.  ...  We first build for each letter aA a list L a of all non isomorphic motifs whose first edge is incident to a vertex of labelā.  ...  The drawback is that not any planar graph has an Hamiltonian path, therefore we could miss some planar maps in our enumeration.  ... 
doi:10.1007/978-3-319-20086-6_18 fatcat:gliczf5xsbevha5dizl4x66qku

Author index to volume 156 (1996)

1996 Discrete Mathematics  
., A reductive technique for enumerating non-isomorphic planar maps (1-3) 197 217 Mishima, M. and M.  ...  Zmazek, On a Vizing-like conjecture for direct product graphs (1 3) 243 246 Kreher, D.L., G.F. Royle and W.D. Wallis, A family of resolvable regular graph designs (Note) , J.H., S.-G. Kim and J.  ... 
doi:10.1016/0012-365x(96)90006-0 fatcat:33jltkh5pbbcha6jvjsffnizqy

Enumerative Formulae for Unrooted Planar Maps: a Pattern

Valery A. Liskovets
2004 Electronic Journal of Combinatorics  
Two simple new enumerative results are deduced for bicolored eulerian maps. A collateral aim is to briefly survey recent and old results of unrooted planar map enumeration.  ...  We present uniformly available simple enumerative formulae for unrooted planar $n$-edge maps (counted up to orientation-preserving isomorphism) of numerous classes including arbitrary, loopless, non-separable  ...  Acknowledgement I am thankful to Timothy Walsh for valuable comments and suggested corrections.  ... 
doi:10.37236/1841 fatcat:dxjknsyr3bbhnd56yqcg2u2ciu

Graph Isomorphism Restricted by Lists [article]

Pavel Klavik, Dušan Knop, Peter Zeman
2016 arXiv   pre-print
A fundamental problem is to construct a combinatorial polynomial-time algorithm for cubic graph isomorphism, avoiding group theory.  ...  The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs G and H, it asks whether they are the same up to a relabeling of vertices.  ...  It seems unlikely that these results will translate to ListIso, but constructing an NP-hardness reduction with bounded eigenvalue multiplicity seems non-trivial.  ... 
arXiv:1607.03918v2 fatcat:rea4gbmazrbebbirsyhndky7mu

Detecting and Counting Small Patterns in Planar Graphs in Subexponential Parameterized Time [article]

Jesper Nederlof
2019 arXiv   pre-print
Generally speaking, our algorithm counts copies of P in time proportional to its number of non-isomorphic separations of order Õ(√(k)).  ...  For counting problems, it was a repeatedly asked open question whether 2^o(k)n^O(1) time algorithms exist that count even special patterns such as independent sets, matchings and paths in planar graphs  ...  In particular Viresh Patel and Guus Regts for an inspiring explanation of [PR17b] , and Viresh Patel for posing Question 8.3 from [PR17b] during a 'training week' of the NETWORKS project (which was the  ... 
arXiv:1904.11285v1 fatcat:ca5c5d3e65fltf7vuv7ezr2x7u

Graph isomorphism, general remarks

Gary L. Miller
1977 Proceedings of the ninth annual ACM symposium on Theory of computing - STOC '77  
These two graphs are not isomorphic; in fact, B is planar while B' is not planar.  ...  Thus many authors have noticed that isomorphism of planar 3-connected graphs is decidable in polynomial time, after finding a polynomial time planar embedding algorithms.  ...  For inclusion a we rely on the fact that the graph is a covering of Heawood's graph. Remark.  ... 
doi:10.1145/800105.803404 dblp:conf/stoc/Miller77 fatcat:qzm3zpv3djhx7pt2hqrqgo5ctq

Graph isomorphism, general remarks

Gary L. Miller
1979 Journal of computer and system sciences (Print)  
These two graphs are not isomorphic; in fact, B is planar while B' is not planar.  ...  Thus many authors have noticed that isomorphism of planar 3-connected graphs is decidable in polynomial time, after finding a polynomial time planar embedding algorithms.  ...  For inclusion a we rely on the fact that the graph is a covering of Heawood's graph. Remark.  ... 
doi:10.1016/0022-0000(79)90043-6 fatcat:l6zqjt26ujfr5lstybn4sjelri

The classification of subfactors with index at most 5 1/4 [article]

Narjess Afzaly, Scott Morrison, David Penneys
2015 arXiv   pre-print
The small index subfactor classification program has been a rich source of interesting quantum symmetries.  ...  Jones, Brendan McKay, and Noah Snyder for helpful conversations. Narjess Afzaly was supported by ARC Discovery Project DP0986827, 'Structure enumeration, applications and analysis'.  ...  Scott Morrison was supported by a Discovery Early Career Research Award DE120100232 and a Discovery Project 'Subfactors and symmetries' DP140100732 from the Australian Research Council. David  ... 
arXiv:1509.00038v1 fatcat:oiuc3dd4mna55cwwq4nkaqvofi

Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs [article]

Jiří Fiala, Pavel Klavík, Jan Kratochvíl, Roman Nedela
2014 arXiv   pre-print
In comparison, testing general graph covers is known to be NP-complete for planar inputs G even for small fixed graphs H such as K_4 or K_5.  ...  When |H|=1, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for |G| = |H| when it becomes the graph isomorphism problem.  ...  Since we have a linear number of non-isomorphic maps, we get the bound 4c·e(H)·v(H), which is O(v 2 (H)).  ... 
arXiv:1402.3774v2 fatcat:l7jlksknavh73fmhzpejmf4pqi

Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs [chapter]

Jiří Fiala, Pavel Klavík, Jan Kratochvíl, Roman Nedela
2014 Lecture Notes in Computer Science  
In comparison, testing general graph covers is known to be NP-complete for planar inputs G even for small fixed graphs H such as K4 or K5.  ...  When |H| = 1, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for |G| = |H| when it becomes the graph isomorphism problem.  ...  Since we have a linear number of non-isomorphic maps, we get the bound 4c·e(H)·v(H), which is O(v 2 (H)).  ... 
doi:10.1007/978-3-662-43948-7_41 fatcat:jsyg36cupbc4jlvbfko5tzgdpa

Page 3430 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
A.| A reductive technique for enumerating non-isomorphic planar maps. (English summary) Discrete Math. 156 (1996), no. 1-3, 197-217.  ...  Summary: “A general method for the counting of unrooted planar maps is proposed.  ... 

Counting unrooted maps on the plane

Valery A. Liskovets, Timothy R. Walsh
2006 Advances in Applied Mathematics  
Previously we obtained formulae for the number of unrooted planar n-edge maps of various classes, including all maps, non-separable maps, eulerian maps and loopless maps.  ...  A planar map is a 2-cell embedding of a connected planar graph, loops and parallel edges allowed, on the sphere. A plane map is a planar map with a distinguished outside ("infinite") face.  ...  The class of all planar maps and of non-separable planar maps are self-dual: for every map in the class, its face-vertex dual is also in the class.  ... 
doi:10.1016/j.aam.2005.03.006 fatcat:d7zfhrrx4baevdm6ctelwqugcm

Subexponential Time Algorithms for Embedding H-Minor Free Graphs

Hans L. Bodlaender, Jesper Nederlof, Tom C. Van Der Zanden, Marc Herbstritt
2016 International Colloquium on Automata, Languages and Programming  
This solves an open problem on the complexity of Subgraph Isomorphism for planar graphs.  ...  This technique seems widely applicable, and it appears that there is a relatively unexplored class of problems that share a similar upper and lower bound.  ...  Traditionally, a canonization is a function that maps non-isomorphic graphs to distinct strings, and isomorphic graphs to identical strings.  ... 
doi:10.4230/lipics.icalp.2016.9 dblp:conf/icalp/BodlaenderNZ16 fatcat:msf7om5zsvht3hflwrweoewfmq
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