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Boundeddegree graphs have arbitrarily large geometric thickness
, 2008
"... The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 200 ..."
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The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists ∆regular graphs with arbitrarily large geometric thickness. In particular, for all ∆ ≥ 9 and for all large n, there exists a ∆regular graph with geometric thickness at least c √ ∆n 1/2−4/∆−ǫ. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th
A note on treepartitionwidth
, 2006
"... Abstract. A treepartition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding ..."
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Abstract. A treepartition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with treewidth k ≥ 3 and maximum degree ∆ ≥ 1 has treepartitionwidth at most 24k∆. We prove that this bound is within a constant factor of optimal. In particular, for all k ≥ 3 and for all sufficiently large ∆, we construct a graph with treewidth k, maximum degree ∆, and treepartitionwidth at least ( 1 8 upper bound to 5
Tree Drawings on the Hexagonal Grid
"... We consider straightline drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges fromeachnodetoitschildrenfromonetofive, andtofivepatterns: straight, Y, ψ, X, and full. The ψ ..."
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We consider straightline drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges fromeachnodetoitschildrenfromonetofive, andtofivepatterns: straight, Y, ψ, X, and full. The ψ–drawings generalize hv or strictly upward drawings to ternary trees. Weshowthatcompleteternarytreeshavea ψ–drawingonasquareofsize O(n 1.262) and general ternary trees can be drawn within O(n 1.631) area. Bothboundsareoptimal.Sub–quadraticboundsarealsoobtainedfor X– pattern drawings of complete tetra trees, and for full–pattern drawings of complete penta trees, which are 4–ary and 5–ary trees. These results parallel and complement the ones of Frati [8] for straight–line orthogonal drawings of ternary trees. Moreover, we provide an algorithm for compacted straight–line drawings of penta trees on the hexagonal grid, such that the direction of the edges from a node to its children is given by our patterns and these edges have the same length. However, drawing trees on a hexagonal grid within a prescribed area or with unit length edges is NP–hard.
MinimumArea Drawings of . . .
, 2011
"... A straightline grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straightline segment. The height, width and area of such a drawing are respectively the height, width and area of the smallest axisali ..."
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A straightline grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straightline segment. The height, width and area of such a drawing are respectively the height, width and area of the smallest axisaligned rectangle on the grid which encloses the drawing. A minimumarea drawing of a plane graph G is a straightline grid drawing of G where the area is the minimum. It is NPcomplete to determine whether a plane graph G has a straightline grid drawing with a given area or not. In this paper we give a polynomialtime algorithm for finding a minimumarea drawing of a plane 3tree. Furthermore, we show a ⌊ 2n 3 −1⌋×2⌈n ⌉ lower bound for the area of a straightline grid drawing of 3 a plane 3tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3
Leftist Canonical Ordering
, 2009
"... Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new appr ..."
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Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering.
www.elsevier.com/locate/comgeo Graph drawings with few slopes ✩
, 2006
"... The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for Δ 5 and all large n, there is a Δregular nvertex graph with slopenumber at least n1− 8+ε Δ+4. This is the best known lower bound on the slopenumber of a gr ..."
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The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for Δ 5 and all large n, there is a Δregular nvertex graph with slopenumber at least n1− 8+ε Δ+4. This is the best known lower bound on the slopenumber of a graph with bounded degree. We prove upper and lower bounds on the slopenumber of complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that the slopenumber of interval graphs, cocomparability graphs, and ATfree graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slopenumber at mostO(logn). Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.