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Lower bounds for the number of edge-crossings over the spine in a topological book embedding of a graph

Hikoe Enomoto, Miki Shimabara Miyauchi, Katsuhiro Ota
1999 Discrete Applied Mathematics  
In a topological book embedding of a graph, the graph is drawn in a topological book by placing the vertices along the spine of the book and drawing the edges in the pages; edges are allowed to cross the  ...  This paper presents lower bounds on the number of edge-crossings over the spine for a variety of graphs. These bounds show that the upper bound O(m log n) is essentially best possible. ? (M.S.  ...  A book embedding with this restriction is a combinatorial book embedding. The pagenumber of a graph is the minimum number of pages in which the graph has a combinatorial book embedding.  ... 
doi:10.1016/s0166-218x(99)00044-x fatcat:4526aholg5bpnesbdzfsp6rmly

Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

Sheren H. Wilar, Benny Pinontoan, Chriestie E.J.C. Montolalu
2021 d CARTESIAN  
The results obtained show that the graph Q_((1,s,b) ) (n) has 10-crossing-critical and book embedding of graph has 4-page book.  ...  A principal tool used in construction of crossing-critical graphs are tiles. In the tile concept, tiles can be arranged by gluing one tile to another in a linear or circular fashion.  ...  The crossing number of a graph is the minimum number of crossing points of edges in a drawing of in the plane. The crossing number of a graph is often denoted as ( ).  ... 
doi:10.35799/dc.9.2.2020.29166 fatcat:fm7evzlezrbjdez267w5mx2y4i

Book embedding of 3-crossing-critical graphs with rational average degree between 3.5 and 4

B Pinontoan, J Titaley, C E J C Montolalu
2019 IOP Conference Series: Materials Science and Engineering  
m, n) on the plane, there are exactly three pairwise edges crossings, and when deleting any edge of the graph, the number of crossings of the remaining graph decreases.  ...  In this paper, we show that the minimal number of pages needed to embed P(m, n) into a book is three.  ...  The crossing number cr(G) of graph G is the minimal number of crossings among all possible drawings of graph G in the plane. The crossing number problem was introduced by Tutte [1] .  ... 
doi:10.1088/1757-899x/567/1/012016 fatcat:maa4wccz4bbafeunoxsnrxlfqe

Various heuristic algorithms to minimise the two-page crossing numbers of graphs

Hongmei He, Ana Sălăgean, Erkki Mäkinen, Imrich Vrt'o
2015 Open Computer Science  
We conjecture that the optimal two-page drawing of a graph represents the planar drawing of the graph.  ...  It is found out that the new heuristics outperform the previously known heuristics and produce good approximations of the planar crossing number for severalwell-known graph families.  ...  Conclusion We designed several heuristics for the two-page book crossing problem by combining two of one-page drawing algorithms (BB+ [12] and our AVSDF+ [13] ) with four twopage algorithms (SLOPE,  ... 
doi:10.1515/comp-2015-0004 fatcat:gmxkbqolvrhkrljq3xpjly435i

Crossing-Optimal Acyclic Hamiltonian Path Completion and Its Application to Upward Topological Book Embeddings [chapter]

Tamara Mchedlidze, Antonios Symvonis
2009 Lecture Notes in Computer Science  
Based on this equivalence we infer for the class of outerplanar triangulated st-digraphs an upward topological 2-page book embedding with minimum number of spine crossings and at most one spine crossing  ...  For the class of planar st-digraphs, we establish an equivalence between the Acyclic-HPCCM problem and the problem of determining an upward 2-page topological book embedding with minimum number of spine  ...  Conclusions -Open Problems We have studied the problem of Acyclic-HPCCM and we have presented a linear time algorithm that computes a crossing-optimal Acyclic HP-completion set for an OT st-digraphs G  ... 
doi:10.1007/978-3-642-00202-1_22 fatcat:6nn4jphp4reorhwnpvjwaqaazi

Page 5176 of Mathematical Reviews Vol. , Issue 96i [page]

1996 Mathematical Reviews  
An upper bound to the crossing number of the complete graph drawn on the pages of a book. International Conference on Graphs and Hypergraphs (Varenna, 1991). J. Combin. Inform.  ...  The crossing number CR*(q, 7) is then defined to be the minimum number of pairwise intersections of edges when K,, is drawn in a book with g pages. .  ... 

On ρ-Constrained Upward Topological Book Embeddings [chapter]

Tamara Mchedlidze, Antonios Symvonis
2010 Lecture Notes in Computer Science  
Then, the unique ρ-constrained upward topological book embedding of G (with a minimum number of spine crossings) can be computed in O(n + α) time, where α is the total number of spine crossings.  ...  Let G be an embedded planar st-digraph and ρ be a topological numbering of G.  ...  In this poster, we present improved results with respect to the number of spine crossings per edge and the time required to compute the book embedding.  ... 
doi:10.1007/978-3-642-11805-0_40 fatcat:7ulewukybvfmzcn67fh5nf5xpu

Experimental Evaluation of Book Drawing Algorithms [article]

Jonathan Klawitter, Tamara Mchedlidze, Martin Nöllenburg
2017 arXiv   pre-print
A k-page book drawing of a graph G=(V,E) consists of a linear ordering of its vertices along a spine and an assignment of each edge to one of the k pages, which are half-planes bounded by the spine.  ...  In this paper, we propose a comprehensive benchmark set of challenging graph classes for book drawing algorithms and provide an extensive experimental study of the performance of existing book drawing  ...  Again, computing the k-page book crossing number, i.e., the minimum number of crossings over all k-page book drawings of a graph, is an NP-hard problem [29, 34] and fixed-parameter tractable algorithms  ... 
arXiv:1708.09221v1 fatcat:2giw66f3xjgw3fal22676t5mva

Embedding a Graph into a d + 1-page Book with m logd n Edge-crossings over the Spine

2005 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
Embedding a Graph into a d + 1-page Book with m log d n Edge-crossings over the Spine Miki MIYAUCHI †a) , Member SUMMARY A topological book embedding of a graph is an embedding in a book that carries the  ...  Enomoto showed that for any graph G having n vertices, there exists a three-page book embedding of G in which each edge crosses the spine log n times.  ...  Theorem 3: There exists a d+1-page topological book embedding of a graph G with m log d n edge-crossings over the spine, where n is the number of vertices and m is the number of edges.  ... 
doi:10.1093/ietfec/e88-a.5.1136 fatcat:rla43itinnfptkgc6z7aycclnm

Optimum embedding of complete graphs in books

Tomasz Bilski
1998 Discrete Mathematics  
Embedding a graph in a book is an arrangement of vertices in a line along the spine of the book and edges on the pages in such a way that edges residing on the same page do not cross.  ...  It is also proved that the set of complete graphs is a proper subset of the set of graphs for which the presented algorithm gives an optimal number of pages.  ...  The width of a book embedding is the maximum number of edges that cross any line perpendicular to the spine of the book on any page.  ... 
doi:10.1016/s0012-365x(97)00131-3 fatcat:m7mungmtvnedrbg3w6wgngh5gi

Fewer bends point-set embedding with mapping

Md. Emran Chowdhury, Muhammad Jawaherul Alam, Md. Saidur Rahman
2010 International Conference on Electrical & Computer Engineering (ICECE 2010)  
Γ is called an upward topological book embedding of with the mapping Φ if the points in are aligned along a straight-line.  ...  Furthermore we also find an upper bound on total number of bends in our drawing, which is, to the best of our knowledge, the first result on the total number of bends for the point-set embedding problem  ...  Fig. 2 . 2 (a) A planar graph , (b) a topological book embedding of with a spine crossing by the edge (2, 5) .  ... 
doi:10.1109/icelce.2010.5700748 fatcat:qcapra5eqvc2dbxfw5q32kpuvi

Page 3740 of Mathematical Reviews Vol. , Issue 96f [page]

1996 Mathematical Reviews  
The book crossing number of a graph G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, so that each edge  ...  Summary: “This paper introduces the book crossing number prob- lem, which can be viewed as a variant of the well-known plane and surface crossing number problem or as a generalization of the book embedding  ... 

Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth [chapter]

Michael J. Bannister, David Eppstein
2014 Lecture Notes in Computer Science  
, and that computing the 2-page crossing number is fixedparameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph.  ...  We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth  ...  This material is based upon work supported by the National Science Foundation under Grant CCF-1228639 and by the Office of Naval Research under Grant No. N00014-08-1-1015.  ... 
doi:10.1007/978-3-662-45803-7_18 fatcat:yol2zw3znzgddpi3ay5avnsmri

Efficient deterministic algorithms for embedding graphs on books [chapter]

Farhad Shahrokhi, Weiping Shi
1996 Lecture Notes in Computer Science  
We derive deterministic polynomial time algorithms for book embedding of a graph G = (V; E), jV j = n and jEj = m.  ...  We then use this algorithm to embed, in polynomial time, any graph G in O( p (G) m) pages, where (G) is the largest minimum degree over all subgraphs of G.  ...  Acknowledgments Research of Weiping Shi was supported by NSF grant MIP-9309120.  ... 
doi:10.1007/3-540-61332-3_149 fatcat:bu5jjouob5brxghev3px53g6tu

Crossing numbers of graphs, lower bound techniques and algorithms: A survey [chapter]

Farhad Shahrokhi, László A. Székely, Imrich Vrt'o
1995 Lecture Notes in Computer Science  
We give a survey of recent techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem.  ...  Our aim is to emphasize the more general resuits or those results which have an algorithmic flavor, including the recent results of the authors.  ...  For a drawing D of G on a k-page book, let vk(D) denote the number of edge crossings in D. Let G1 =< V1,E1 > and G~ =< V2,E2 > be two undirected graphs, IV1[ < [l/~[.  ... 
doi:10.1007/3-540-58950-3_364 fatcat:tv2kip32lfartiaifhul6vlwui
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