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A reductive technique for enumerating non-isomorphic planar maps

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. ...

##
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Efficient Generation of Stable Planar Cages for Chemistry
[article]

2015
*
arXiv
*
pre-print

In this paper we describe an algorithm which generates all colored

arXiv:1503.06610v1
fatcat:ywb4wcysrbdnbhuzo7t3ihyyf4
*planar**maps*with*a*good minimum sparsity from simple motifs and rules to connect them. ... We first build*for*each letter*a*∈*A**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*. ...##
###
Efficient Generation of Stable Planar Cages for Chemistry
[chapter]

2015
*
Lecture Notes in Computer Science
*

In this paper we describe an algorithm which generates all colored

doi:10.1007/978-3-319-20086-6_18
fatcat:gliczf5xsbevha5dizl4x66qku
*planar**maps*with*a*good minimum sparsity from simple motifs and rules to connect them. ... We first build*for*each letter*a*∈*A**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*. ...##
###
Author index to volume 156 (1996)

1996
*
Discrete Mathematics
*

.,

doi:10.1016/0012-365x(96)90006-0
fatcat:33jltkh5pbbcha6jvjsffnizqy
*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. ...##
###
Enumerative Formulae for Unrooted Planar Maps: a Pattern

2004
*
Electronic Journal of Combinatorics
*

Two simple new enumerative results are deduced

doi:10.37236/1841
fatcat:dxjknsyr3bbhnd56yqcg2u2ciu
*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. ...##
###
Graph Isomorphism Restricted by Lists
[article]

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. ...

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

2019
*
arXiv
*
pre-print

Generally speaking, our algorithm counts copies of P in time proportional to its number of

arXiv:1904.11285v1
fatcat:ca5c5d3e65fltf7vuv7ezr2x7u
*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 ...##
###
Graph isomorphism, general remarks

1977
*
Proceedings of the ninth annual ACM symposium on Theory of computing - STOC '77
*

These two graphs are not

doi:10.1145/800105.803404
dblp:conf/stoc/Miller77
fatcat:qzm3zpv3djhx7pt2hqrqgo5ctq
*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. ...##
###
Graph isomorphism, general remarks

1979
*
Journal of computer and system sciences (Print)
*

These two graphs are not

doi:10.1016/0022-0000(79)90043-6
fatcat:l6zqjt26ujfr5lstybn4sjelri
*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. ...##
###
The classification of subfactors with index at most 5 1/4
[article]

2015
*
arXiv
*
pre-print

The small index subfactor classification program has been

arXiv:1509.00038v1
fatcat:oiuc3dd4mna55cwwq4nkaqvofi
*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 ...##
###
Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs
[article]

2014
*
arXiv
*
pre-print

In comparison, testing general graph covers is known to be NP-complete

arXiv:1402.3774v2
fatcat:l7jlksknavh73fmhzpejmf4pqi
*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)). ...##
###
Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

In comparison, testing general graph covers is known to be NP-complete

doi:10.1007/978-3-662-43948-7_41
fatcat:jsyg36cupbc4jlvbfko5tzgdpa
*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)). ...##
###
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

2006
*
Advances in Applied Mathematics
*

Previously we obtained formulae

doi:10.1016/j.aam.2005.03.006
fatcat:d7zfhrrx4baevdm6ctelwqugcm
*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. ...##
###
Subexponential Time Algorithms for Embedding H-Minor Free Graphs

2016
*
International Colloquium on Automata, Languages and Programming
*

This solves an open problem on the complexity of Subgraph

doi:10.4230/lipics.icalp.2016.9
dblp:conf/icalp/BodlaenderNZ16
fatcat:msf7om5zsvht3hflwrweoewfmq
*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. ...
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