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Drawing planar graphs of bounded degree with few slopes
 SIAM J. Discrete Math
"... We settle a problem of Dujmovi¢, Eppstein, Suderman, and Wood by showing that there exists a function f with the property that every planar graph G with maximum degree d admits a drawing with noncrossing straightline edges, using at most f(d) dierent slopes. If we allow the edges to be represented ..."
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Cited by 8 (0 self)
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We settle a problem of Dujmovi¢, Eppstein, Suderman, and Wood by showing that there exists a function f with the property that every planar graph G with maximum degree d admits a drawing with noncrossing straightline edges, using at most f(d) dierent slopes. If we allow the edges to be represented by polygonal paths with one bend, then 2d slopes suce. Allowing two bends per edge, every planar graph with maximum degree d ≥ 3 can be drawn using segments of at most dd/2e dierent slopes. There is only one exception: the graph formed by the edges of an octahedron is 4regular, yet it requires 3 slopes. These bounds cannot be improved. 1
Decomposition of Geometric Set Systems and Graphs
"... We study two decomposition problems in combinatorial geometry. The rst part of the thesis deals with the decomposition of multiple coverings of the plane. We say that a planar set is coverdecomposable if there is a constant m such that any mfold covering of the plane with its translates is decomp ..."
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Cited by 7 (6 self)
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We study two decomposition problems in combinatorial geometry. The rst part of the thesis deals with the decomposition of multiple coverings of the plane. We say that a planar set is coverdecomposable if there is a constant m such that any mfold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is coverdecomposable. We verify his conjecture for polygons. Moreover, if m is large enough, depending on k and the polygon, we prove that any mfold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in three dimensions, for any polyhedron and any m we construct an mfold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not coverdecomposable. We start the rst part with a detailed survey of all results on the coverdecomposability of polygons. The second part of the thesis investigates another geometric partition problem, related to planar representation of graphs. Wade and Chu dened the slope number of a graph G as the smallest number s with the property that G has a straightline drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to innity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only ve slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, dened only for planar graphs as the smallest number s with the property that the graph has a straightline drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.
Regular paper Communicated by:
, 2015
"... In octilinear drawings of planar graphs, every edge is drawn as a sequence of horizontal, vertical and diagonal (45◦) line segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A kplanar graph is a planar graph in which each vertex has degree a ..."
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In octilinear drawings of planar graphs, every edge is drawn as a sequence of horizontal, vertical and diagonal (45◦) line segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A kplanar graph is a planar graph in which each vertex has degree at most k. In particular, we prove that every 4planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size O(n2) × O(n). For 5planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in superpolynomial area. However, for 6planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge for some edges. Submitted:
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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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 degree ∆ admits an outer 1planar straightline drawing that uses O(∆) different slopes, which generalizes a previous result by Knauer et al. about the outerplanar slope number of outerplanar graphs [18]. We also show that O(∆2) slopes suffice to construct a crossingfree straightline drawing of G; 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].
Planar Octilinear Drawings with One Bend Per Edge
"... In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal (45◦) linesegments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A kplanar graph is a planar graph in which each vertex ..."
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In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal (45◦) linesegments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A kplanar graph is a planar graph in which each vertex has degree less or equal to k. In particular, we prove that every 4planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size O(n2)×O(n). For 5planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in superpolynomial area. However, for 6planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge. 1