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Chordal embeddings of planar graphs

V. Bouchitté, F. Mazoit, I. Todinca
2003 Discrete Mathematics  
Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual di er by at most one.  ...  We give here a much shorter proof of this result.  ...  First, we recall how to obtain minimal chordal embeddings of graphs by completing some families of minimal separators.  ... 
doi:10.1016/s0012-365x(03)00230-9 fatcat:np5dp4zedfg2bivcpqw77ziy4m

Planar domination graphs

Elaine M. Eschen, William F. Klostermeyer, R. Sritharan
2003 Discrete Mathematics  
In this paper, it is shown that the class of planar domination graphs is equivalent to the class of planar weakly chordal graphs, and thus, can be recognized in polynomial time.  ...  A graph G is a domination graph if each induced subgraph of G has a pair of vertices such that the open neighborhood of one is contained in the closed neighborhood of the other in the subgraph.  ...  Acknowledgements The authors wish to thank the referees for their careful reading of the paper.  ... 
doi:10.1016/s0012-365x(02)00684-2 fatcat:jeupomwwpvfhvbxnthsyebbvlm

Planar Hamiltonian chordal graphs are cycle extendable

Tao Jiang
2002 Discrete Mathematics  
Hendry (Discrete Math. 85 (1990) 59) asked if every Hamiltonian chordal graph is cycle extendable. We prove that every planar Hamiltonian chordal graph is cycle extendable.  ...  A graph G is cycle extendable if G contains at least one cycle and every non-Hamiltonian cycle in G is extendable.  ...  It follows that C contains more edges of C than H does. Theorem 2 . 2 Every planar Hamiltonian chordal graph is cycle extendable. Proof. Let G be a planar Hamiltonian chordal graph.  ... 
doi:10.1016/s0012-365x(02)00505-8 fatcat:5wj4zroievfzdplvm63rdmqsje

Outerspatial 2-complexes: Extending the class of outerplanar graphs to three dimensions [article]

Johannes Carmesin, Tsvetomir Mihaylov
2021 arXiv   pre-print
This is applied to nested plane embeddings of graphs; that is, plane embeddings constrained by conditions placed on a set of cycles of the graph.  ...  We introduce the class of outerspatial 2-complexes as the natural generalisation of the class of outerplanar graphs to three dimensions.  ...  As remarked after the statement of Lemma 4.10, this completes the proof of Theorem 3.7, which implies Theorem 1.1 (as shown in Observation 3.8).  ... 
arXiv:2103.15404v2 fatcat:seqy7gmupvgslp4hntw5ujkaru

The existence of homeomorphic subgraphs in chordal graphs

C.R. Subramanian, C.E. Veni Madhavan
1997 Applied Mathematics Letters  
Using these results, we develop a simple linear time algorithm for testing planarity of chordal graphs.  ...  We establish conditions for the existence, in a chordal graph, of subgraphs homeomorphic to Kn (n >_ 3), Km,n (m, n >_ 2), and wheels Wr (r _> 3).  ...  Efficient Planarity Testing Algorithm for Chordal Graphs Using the characterization of planar chordal graphs given in Theorem 2.3, we develop an alternative linear time planarity testing algorithm for  ... 
doi:10.1016/s0893-9659(97)00027-x fatcat:q4663fbr2vdzhkgsbj3u73gfoq

Survey on Plane 3-Tree with Nearest Neighbor Interchanges and Chordal Bipartite Graph

Tamanna Afroze
2017 International Journal of Computer Applications  
In this survey paper, I want to present the plane 3-tree's concept with an interesting branch of another topic of graph theory, that is, chordal bipartite graph.  ...  Throughout the development of this survey paper, we will present definitions of chordal graph, bipartite graph, tree, plane 3-tree and different operations of tree architecture.  ...  Thus in any embedding of a maximal planar graph G with n 3, the boundary of every face of G is a triangle, and hence the embedding is often called a triangulated plane graph.  ... 
doi:10.5120/ijca2017913298 fatcat:lpdor55livdl3osuyac4udj7ru

Halin graphs are 3-vertex-colorable except even wheels [article]

A. Kapanowski, A. Krawczyk
2019 arXiv   pre-print
A Halin graph is a graph obtained by embedding a tree having no nodes of degree two in the plane, and then adding a cycle to join the leaves of the tree in such a way that the resulting graph is planar  ...  We show that all Halin graphs are 3-vertex-colorable except even wheels. We also show how to find the perfect elimination ordering of a chordal completion for a given Halin graph.  ...  One can find a planar embedding of a given graph and then choose the proper outer face with m − n + 1 edges [8] .  ... 
arXiv:1903.02904v1 fatcat:hve6e6dqa5c45nvjxz3mo4rlhm

Page 7328 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
Although planar graphs are usually defined by embeddings into the two-dimensional real space, this definition can hardly be used for actually developing a formal theory of planar graphs.  ...  The centers of connected graphs having some property P, where P is acyclic, unicyclic, maximal outer planar, chordal, etc., have been well studied by many authors.  ... 

Macroscopic network circulation for planar graphs [article]

Fariba Ariaei, Zahra Askarzadeh, Yongxin Chen, Tryphon T. Georgiou
2020 arXiv   pre-print
While we introduce and propose these notions in a general setting, in this paper, we only work out the case of planar graphs.  ...  Beyond planar graphs, in the general case, the problem to determine global circulation remains at present a combinatorial problem.  ...  This way, we generate a planar chordal graph such that every chordless cycle subgraph is a triangle.  ... 
arXiv:2004.02053v2 fatcat:da4ygsxoina2pamyuwt4p4gofm

The pagenumber of k-trees is O(k)

Joseph L. Ganley, Lenwood S. Heath
2001 Discrete Applied Mathematics  
The pagenumber of a graph is the minimum number of pages in a valid book embedding. In this paper, it is proven that the pagenumber of a k-tree is at most k + 1.  ...  A book embedding of a graph maps the vertices onto a line along the spine of the book and assigns the edges to pages of the book such that no two edges on the same page cross.  ...  Thanks also to Hans Bodlaender for some helpful discussions on classes of graphs with bounded treewidth.  ... 
doi:10.1016/s0166-218x(00)00178-5 fatcat:bopstnc3sbbqfesjvwbw2kvq5m

Schnyder woods for higher genus triangulated surfaces

Luca Castelli Aleardi, Eric Fusy, Thomas Lewiner
2008 Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08  
case of graphs embedded on surfaces of arbitrary genus.  ...  Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general  ...  Colin de 18 Verdière for pointing out some useful topological properties of graphs on surfaces. We are extremely grateful to O. Bernardi and G.  ... 
doi:10.1145/1377676.1377730 dblp:conf/compgeom/AleardiFL08 fatcat:be5hcdumdva3bkhw6ncibovlmi

Page 2013 of Mathematical Reviews Vol. , Issue 96d [page]

1996 Mathematical Reviews  
In each iteration of the algorithm, either the number of edges decreases or a vertex of the planar graph or its dual graph is deleted. A. D.  ...  A graph is chordal bipartite if it is bipartite and every cycle of length at least six has a chord.  ... 

Page 51 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
In fact, Walter, Gavril, and Buneman have shown that G is a chordal graph iff G is the intersection graph of a family of subtrees of a tree. So DV, RDV and UV graphs are subclasses of chordal graphs.  ...  It is shown in the paper that if a group possesses a 3-connected planar Cayley graph, then the graph may be embedded in the sphere in such a way that the group action on the Cayley graph can be realized  ... 

Toughness and Hamiltonicity of a class of planar graphs

T. Gerlach
2004 Discrete Mathematics  
A graph G is called chordal if every cycle of G of length at least four has a chord. By a theorem of B ohme, Harant and Tkà aÄ c more than 1-tough chordal planar graphs are Hamiltonian.  ...  We prove that this is even true for more than 1-tough planar graphs under the weaker assumption that separating cycles of length at least four have chords.  ...  Every chordal planar graph with toughness greater than one is Hamiltonian, while the shortness exponent of the class of all 1-tough chordal planar graphs is at most log 8 log 9 .  ... 
doi:10.1016/j.disc.2003.11.046 fatcat:utdlld4j7fbphi4w456es5ufgu

Graphs with Plane Outside-Obstacle Representations [article]

Alexander Koch and Marcus Krug and Ignaz Rutter
2013 arXiv   pre-print
As a side result, we show that the plane vertex--polygon visibility graphs are exactly the maximal outerplanar graphs and that every chordal outerplanar graph has an outside-obstacle representation.  ...  An obstacle representation of a graph consists of a set of polygonal obstacles and a distinct point for each vertex such that two points see each other if and only if the corresponding vertices are adjacent  ...  Acknowledgments Part of this work has been done while Alexander Koch participated in the academic exchange program of Tōhoku University and KIT. AK thanks Prof. Dorothea Wagner and Prof.  ... 
arXiv:1306.2978v1 fatcat:622wiotbjzcmretmcejiu2k2tq
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