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The Planar Slope Number of Planar Partial 3Trees of Bounded Degree
, 2009
"... It is known that every planar graph has a planar embedding where edges are represented by noncrossing straightline segments. We study the planar slope number, i.e., the minimum number of distinct edgeslopes in such a drawing of a planar graph with maximum degree Δ. We show that the planar slope ..."
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Cited by 5 (0 self)
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number of every seriesparallel graph of maximum degree three is three. We also show that the planar slope number of every planar partial 3tree and also every plane partial 3tree is at most 2O(Δ). In particular, we answer the question of Dujmovic ́ et al. [Computational Geometry 38 (3), pp. 194
Drawing outer 1planar graphs with few slopes
, 2015
"... A graph is outer 1planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3trees. We show that an outer 1planar graph G of bounded d ..."
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; the best known upper bound on the planar slope number of planar partial 3trees of bounded degree ∆ is O(∆5) as proved by Jeĺınek et al. [16].
On the number of planar orientations with prescribed degrees
, 2008
"... We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α: V → N unifies many diffe ..."
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Cited by 9 (3 self)
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.37 n Schnyder woods, 3connected planar maps with 3.209 n Schnyder woods and inner triangulations with 2.91 n bipolar orientations. These lower bounds are accompanied by upper bounds of 3.56 n, 8 n and 3.97 n respectively. We also show that for any planar map M and any α the number of α
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
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Cited by 14 (1 self)
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and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Really straight drawings II: Nonplanar graphs
, 2005
"... We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings ..."
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Cited by 1 (1 self)
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We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have
Drawing Planar Graphs with Reduced Height
"... Abstract. A straightline (respectively, polyline) drawing Γ of a planar graph G on a set Lk of k parallel lines is a planar drawing that maps each vertex of G to a distinct point on Lk and each edge of G to a straight line segment (respectively, a polygonal chain with the bends on Lk) between its e ..."
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endpoints. The height of Γ is k, i.e., the number of lines used in the drawing. In this paper we compute new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every nvertex planar graph with maximum degree ∆, having a simple cycle separator
Planar LShaped Point Set Embeddings of Trees *
"... Abstract In this paper we consider planar Lshaped embeddings of trees in point sets, that is, planar drawings where the vertices are mapped to a subset of the given points and where every edge consists of two axisaligned line segments. We investigate the minimum number m, such that any n vertex t ..."
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tree with maximum degree 4 admits a planar Lshaped embedding in any point set of size m. First we give an upper bound O(n c ) with c = log 2 3 ≈ 1.585 for the general case, and thus answer the question by Di Giacomo et al. [4] whether a subquadratic upper bound exists. Then we introduce the saturation
A Note on MinimumSegment Drawings of Planar Graphs
"... A straightline drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straightline drawing is a maximal set of edges that form a straight line segment. A minimumsegment d ..."
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Cited by 1 (1 self)
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that the problem of deciding whether a given partial drawing of G can be extended to a straightline drawing with at most k segments is NPcomplete, even when G is an outerplanar graph. Finally, we investigate a worstcase lower bound on the number of segments required by straightline drawings of arbitrary
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 ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a ..."
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Cited by 14 (3 self)
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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 AT
External memory bfs on undirected graphs with bounded degree
 In Proceedings of SODA’2001
, 2001
"... We give the first external memory algorithm for breadthfirst search (BFS) which achieves o(n) I/Os on arbitrary undirected graphs with n nodes and maximum node degree d. Let M and B> d denote the main memory size and block size, respectively. Using Sort(x) = O( ~.IOgM/B ~), our algorithm needs ..."
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Cited by 15 (4 self)
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undirected graphs still require f~(n) I/Os, even if the graphs are planar and/or have bounded node degrees: O(n+~.Sor t (n) ) I/Os for BFS [4], O(n+~log ~) I/Os for SSSP [3], and O(min{~~. ~ +n, (n+~). log ~}) I/Os for DFS [6]. Better algorithms are known for special graph classes, see [6
Results 1  10
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64