362 research outputs found

    Simplicial families of drawings

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    Cubic Partial Cubes from Simplicial Arrangements

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    We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure

    Graph Treewidth and Geometric Thickness Parameters

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    Consider a drawing of a graph GG in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of GG, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth kk, the maximum thickness and the maximum geometric thickness both equal k/2\lceil{k/2}\rceil. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth kk, the maximum book thickness equals kk if k2k \leq 2 and equals k+1k+1 if k3k \geq 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

    Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

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    In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangulations include the Brieskorn homology spheres Σ(p,q,r)\Sigma (p,q,r), the lens spaces L(q,1)\operatorname{L} (q,1) and, as a limit case, (S2×S1)#(p1)(q1)(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}.Comment: 28 pages, 9 figures. Minor update. To appear in Israel Journal of Mathematic

    Arc Operads and Arc Algebras

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    Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy, and it similarly acts on the loop space of any topological space. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper15.abs.htm
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