7 research outputs found
Relating Graph Thickness to Planar Layers and Bend Complexity
The thickness of a graph with vertices is the minimum number of
planar subgraphs of whose union is . A polyline drawing of in
is a drawing of , where each vertex is mapped to a
point and each edge is mapped to a polygonal chain. Bend and layer complexities
are two important aesthetics of such a drawing. The bend complexity of
is the maximum number of bends per edge in , and the layer complexity
of is the minimum integer such that the set of polygonal chains in
can be partitioned into disjoint sets, where each set corresponds
to a planar polyline drawing. Let be a graph of thickness . By
F\'{a}ry's theorem, if , then can be drawn on a single layer with bend
complexity . A few extensions to higher thickness are known, e.g., if
(resp., ), then can be drawn on layers with bend complexity 2
(resp., ). However, allowing a higher number of layers may reduce the
bend complexity, e.g., complete graphs require layers to be drawn
using 0 bends per edge.
In this paper we present an elegant extension of F\'{a}ry's theorem to draw
graphs of thickness . We first prove that thickness- graphs can be
drawn on layers with bends per edge. We then develop another
technique to draw thickness- graphs on layers with bend complexity,
i.e., , where . Previously, the bend complexity was not known to be sublinear for
. Finally, we show that graphs with linear arboricity can be drawn on
layers with bend complexity .Comment: A preliminary version appeared at the 43rd International Colloquium
on Automata, Languages and Programming (ICALP 2016
Geometric Thickness in a Grid
Th geometric thmetric of agraph is th minimum number of layerssuch thh th graph can be drawn in th planewith edges asstraigh-VV=V segments, andwith each edge assigned to a layer soth9 no two edges in th same layer cross. We consider a variation onthV thV= in whRE each edge is allowed one bend. We provethv th vertices of an n-vertex m-edgegraph grid and th edges assigned to O( # m) layers, sothE each edge is drawnwith at most one bend and no two edges on th same layer cross.Th proof is a 2-dimensional generalisation of athzCBE of Malitz (J. Algorith17 (1) (1994) 71--84) on book embeddings. We obtain a Las Vegasalgorith to computeth probability)