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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]

Parveen, Jitender Kumar
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.  ... 
arXiv:2207.04641v1 fatcat:crhjdl7cebhv3fphqo7jfv5sym

Keeping Avoider's graph almost acyclic [article]

Dennis Clemens, Julia Ehrenmüller, Yury Person, Tuan Tran
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.  ... 
arXiv:1403.1482v1 fatcat:6q2xe4szuffy3ptzinvbrq4riu

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

Rahul Arora and Ashu Gupta and Rohit Gurjar and Raghunath Tewari
2014 arXiv   pre-print
Such a construction was known previously for 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.  ... 
arXiv:1411.7614v1 fatcat:2ngy3s2b2nhg5kfnw3l2xrccca

Space complexity: what makes planar graphs special?

Samir Datta, Raghav Kulkarni
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 ?  ... 
dblp:journals/eatcs/DattaK13 fatcat:errya5hbabeg5bh2qph5yobtiy

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

Mustapha Kchikech, Olivier Togni
2006 Discrete Mathematics & Theoretical Computer Science  
We first propose an 11/6-approximation algorithm for multicoloring any weighted 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.  ... 
doi:10.46298/dmtcs.371 fatcat:w4czre2fgvg5dcp2r6h42okx5m

Localized topology control for unicast and broadcast in wireless ad hoc networks

Wen-Zhan Song, Xiang-Yang Li, O. Frieder, Weu Zhao Wang
2006 IEEE Transactions on Parallel and Distributed Systems  
We theoretically prove that 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).  ... 
doi:10.1109/tpds.2006.53 fatcat:2klyyw3dmjhrhfp754amo527ou

Ends of branching random walks on planar hyperbolic Cayley graphs [article]

Lorenz A. Gilch, Sebastian Müller
2014 arXiv   pre-print
We prove that 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.  ... 
arXiv:1409.3443v1 fatcat:fgivf7hyoze3lk3thlztbj4h3m

A Systematic Approach to Group Properties Using its Geometric Structure

Muhammed Bello, Nor Muhainiah Mohd Ali, Nurfarah Zulkifli
2020 European Journal of Pure and Applied Mathematics  
We give 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.  ... 
doi:10.29020/nybg.ejpam.v13i1.3587 fatcat:uyswrt2i3jejhe6r42jr5vfojm

Enumeration and Asymptotic Properties of Unlabeled Outerplanar Graphs

Manuel Bodirsky, Éric Fusy, Mihyun Kang, Stefan Vigerske
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] .  ... 
doi:10.37236/984 fatcat:jd7sgnqbi5g7zpayq7yoboc7vi

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

Shangjing Hao, Guo Zhong, Xuanlong Ma
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 ]).  ... 
doi:10.7546/crabs.2022.03.03 fatcat:l2vr3tovsbaq7a2gmpwr2cetsi

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

Yusuke Kobayashi, Ken-ichi Kawarabayashi
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  ... 
doi:10.1137/1.9781611973068.124 fatcat:w7ovhq54lzfajij577frtpc5l4

The Cayley graphs of Coxeter and Artin groups

Carl Droms, Herman Servatius
1993 Proceedings of the American Mathematical Society  
We obtain a complete classification 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.  ... 
doi:10.1090/s0002-9939-1993-1140668-4 fatcat:eah2sc23qzgwjbbltujz3q2ara

Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs [chapter]

Xuemin Lin, Peter Eades
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.  ... 
doi:10.1007/3-540-48686-0_9 fatcat:u3mbk5b4ejegpabyxbttnnhgmi
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