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Page 563 of Mathematical Reviews Vol. 47, Issue 3
[page]

1974
*
Mathematical Reviews
*

*In*this paper

*the*concept

*of*unique embeddability is intro- duced

*in*terms

*of*special types

*of*isomorphisms between any two embeddings

*of*a

*planar*

*graph*. ...

*In*this manner,

*the*term ‘uniquely embeddable

*planar*

*graph*’ was introduced. ...

##
###
The complement of enhanced power graph of a finite group
[article]

2022
*
arXiv
*
pre-print

*In*this article, we give an affirmative answer

*of*

*the*question posed by Cameron [6] which states that: Is it true that

*the*complement

*of*

*the*enhanced

*power*

*graph*𝒫̅_̅E̅(̅G̅)̅

*of*a non-cyclic group G has ... only

*one*connected component apart from

*isolated*vertices? ...

*In*section 3, we have proved that

*the*complement

*of*

*the*enhanced

*power*

*graph*

*of*a finite group has just

*one*connected components apart from

*isolated*vertices. ...

##
###
Keeping Avoider's graph almost acyclic
[article]

2014
*
arXiv
*
pre-print

*In*particular, we show that Avoider can keep his

*graph*being a forest for every but maybe

*the*last round

*of*

*the*game if b ≥ 200 n n. ... Moreover, we give a slight improvement for

*the*lower bound

*in*

*the*non-

*planarity*game. ... We are grateful to Miloš Stojaković for helping us to further improve our bounds

*in*Theorem 1.1 and Theorem 1.2. ...

##
###
Derandomizing Isolation Lemma for K_3,3-free and K_5-free Bipartite Graphs
[article]

2014
*
arXiv
*
pre-print

Such a construction was known previously for

arXiv:1411.7614v1
fatcat:2ngy3s2b2nhg5kfnw3l2xrccca
*planar*bipartite*graphs*. Our result implies that*the*perfect matching problem for K_3,3-free and K_5-free bipartite*graphs*is*in*SPL. ...*The**Isolation*Lemma states that giving a random weight assignment to*the*edges*of*a*graph*, ensures that it has a unique minimum weight perfect matching, with a good probability. ... Subsequently, Search-PM was also shown to be*in*RNC [KUW86, MVV87] .*The*solution*of*Mulmuley et al. [MVV87] was based*on**the**powerful*idea*of**Isolation*Lemma. ...##
###
Space complexity: what makes planar graphs special?

2013
*
Bulletin of the European Association for Theoretical Computer Science
*

*In*this timely and well-written article, Samir Datta and Raghav Kulkarni focus

*on*

*planar*

*graphs*

*in*

*the*context

*of*space complexity. ... They discuss several properties

*of*

*planar*

*graphs*and their precise role

*in*

*the*design

*of*space efficient algorithms. ...

*Isolation*

*in*Non-bipartite

*Planar*

*Graphs*Can

*one*extend

*the*method

*of*non-vanishing circulation

*of*Datta, Kulkarni, and Roy [11] to non-bipartite

*planar*

*graphs*? ...

##
###
Approximation Algorithms for Multicoloring Planar Graphs and Powers of Square and Triangular Meshes

2006
*
Discrete Mathematics & Theoretical Computer Science
*

We first propose an 11/6-approximation algorithm for multicoloring any weighted

doi:10.46298/dmtcs.371
fatcat:w4czre2fgvg5dcp2r6h42okx5m
*planar**graph*. We then study*the*multicoloring problem*on**powers**of*square and triangular meshes. ...*The*main focus*of*this work is to obtain upper bounds*on**the*weighted chromatic number*of*some classes*of**graphs**in*terms*of**the*weighted clique number. ... Acknowledgements*The*authors would like to thank*the*anonymous referees for many helpful comments which greatly improved*the*presentation*of**the*paper. ...##
###
Localized topology control for unicast and broadcast in wireless ad hoc networks

2006
*
IEEE Transactions on Parallel and Distributed Systems
*

We theoretically prove that

doi:10.1109/tpds.2006.53
fatcat:2klyyw3dmjhrhfp754amo527ou
*the*proposed topology is*planar*, which meets*the*requirement*of*certain localized routing methods to guarantee packet delivery; it is*power*efficient for unicast-*the*energy ... Moreover, by assuming that*the*node ID and its position can be represented*in*O(log n) bits for a wireless network*of*n nodes,*the*total number*of*messages by our methods is*in**the*range*of*[5n, 13n], ... Fig. 1 . 1 Several*planar**power*spanners*on**the*UDG shown*in*(a). ...##
###
Ends of branching random walks on planar hyperbolic Cayley graphs
[article]

2014
*
arXiv
*
pre-print

We prove that

arXiv:1409.3443v1
fatcat:fgivf7hyoze3lk3thlztbj4h3m
*the*trace*of*a transient branching random walk*on*a*planar*hyperbolic Cayley*graph*has a.s. continuum many ends and no*isolated*end. ... Acknowledgment*The*authors thank Elisabetta Candellero and Matthew Roberts for comments*on*a first version*of*this note.*The*research was supported by*the*exchange programme Amadeus-Amadée 31473TF. ...*The*trace*of*a transient UBRW*on*a*planar*hyperbolic Cayley*graph*has a.s. no*isolated*end. ...##
###
A Systematic Approach to Group Properties Using its Geometric Structure

2020
*
European Journal of Pure and Applied Mathematics
*

We give

doi:10.29020/nybg.ejpam.v13i1.3587
fatcat:uyswrt2i3jejhe6r42jr5vfojm
*the*general presentation for*the**graph**on*dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups*in*terms*of**the*order product prime*graphs*as*one**of*connected, ...*The**graph**on*dihedral groups is proven to be regular and complete only if*the*degree*of**the*corresponding group is even prime*power*and connected for all prime*power*degree. ... Acknowledgements*The*first author would like to thank Federal University*of*Kashere (FUK) for their total support. ...##
###
Enumeration and Asymptotic Properties of Unlabeled Outerplanar Graphs

2007
*
Electronic Journal of Combinatorics
*

*The*exact number $g_{n}$

*of*unlabeled outerplanar

*graphs*

*on*$n$ vertices can be computed

*in*polynomial time, and $g_{n}$ is asymptotically $g\, n^{-5/2}\rho^{-n}$, where $g\approx0.00909941$ and $\rho^ ... We determine

*the*exact and asymptotic number

*of*unlabeled outerplanar

*graphs*. ...

*The*number

*of*three-connected

*planar*maps is related to

*the*number

*of*three-connected

*planar*

*graphs*[25, 35] , since a three-connected

*planar*

*graph*has a unique embedding

*on*

*the*sphere [40] . ...

##
###
Page 1399 of Mathematical Reviews Vol. , Issue 82d
[page]

1982
*
Mathematical Reviews
*

*In*this terminology he gives three theorems concerning subgraphs consisting only

*of*

*isolated*points and concerning

*the*chromatic number

*of*

*the*given

*graph*. H. ...

*The*theorem allows

*one*to recognize that G is

*planar*, and, because

*of*

*the*order and size, maximal

*planar*. ...

##
###
Notes on the Co-prime Order Graph of a Group

2022
*
Comptes Rendus de l'Academie Bulgare des Sciences / Proceedings of the Bulgarian Academy of Sciences
*

*The*co-prime order

*graph*

*of*a group $$G$$ is

*the*

*graph*with vertex set $$G$$, and two distinct elements $$x,y\

*in*G$$ are adjacent if gcd$$(o(x),o(y))$$ is either $$1$$ or a prime, where $$o(x)$$ and $$ ...

*In*this paper, we characterize finite groups whose co-prime order

*graphs*are complete and classify finite groups whose co-prime order

*graphs*are

*planar*, which generalizes some results by Banerjee [3]. ...

*The*

*power*

*graph*

*of*a group G has

*the*vertex set G and two distinct elements are adjacent if

*one*is a

*power*

*of*

*the*other (cf. [ 1 ]). ...

##
###
Algorithms for Finding an Induced Cycle in Planar Graphs and Bounded Genus Graphs
[chapter]

2009
*
Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
*

*In*this paper, we consider

*the*problem

*of*finding an induced cycle passing through k given vertices, which we call

*the*induced cycle problem. ... For any instance

*of*

*the*ICP

*in*a

*planar*

*graph*with k terminals every (22k + 2)-

*isolated*vertex is deletable. This is

*one*

*of*our main contributions, and we give a sketch

*of*

*the*proof. ... Since we only consider

*planar*

*graphs*and

*graphs*embedded

*in*a fixed surface, we do not need

*the*full

*power*

*of*Robertson and Seymour's proof [24] (but we still need a deep topological result

*in*

*Graph*...

##
###
The Cayley graphs of Coxeter and Artin groups

1993
*
Proceedings of the American Mathematical Society
*

We obtain a complete classification

doi:10.1090/s0002-9939-1993-1140668-4
fatcat:eah2sc23qzgwjbbltujz3q2ara
*of**the*Coxeter and Artin groups whose Cayley*graphs*with respect to*the*standard presentations are*planar*. ... We also classify those whose Cayley*graphs*have*planar*embeddings*in*which*the*vertices have no accumulation points. ... A*graph*Y is called outer*planar*if it can be drawn*in**the*plane*in*such a way that all vertices lie*on**the*boundary*of*a single face. ...##
###
Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

*In*this paper, we shall investigate

*the*computational complexity

*of*constructing minimum area grid visibility representations

*of*hierarchically

*planar*

*graphs*. ... This implies that

*the*area minimization problem is polynomial time solvable restricted to

*the*class

*of*

*graphs*whose

*planar*embeddings are unique. ...

*In*this paper we are focused only

*on*a \visibility representation"

*of*a hierarchically

*planar*

*graph*. ...

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