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On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
Abstract

Cited by 31 (19 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
Geometric Thickness in a Grid
 Discrete Mathematics
, 2001
"... The geometric thickness of a graph is the minimum number of layers such that the graph can be drawn in the plane with edges as straightline segments, and with each edge assigned to a layer so that no two edges on the same layer cross. We consider a variation on this theme in which each edge is allo ..."
Abstract

Cited by 2 (0 self)
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The geometric thickness of a graph is the minimum number of layers such that the graph can be drawn in the plane with edges as straightline segments, and with each edge assigned to a layer so that no two edges on the same layer cross. We consider a variation on this theme in which each edge is allowed one bend. We prove that the vertices of an nvertex medge graph can be positioned in a $$ grid and the edges assigned to $$ layers, so that each edge is drawn with at most one bend and no two edges on the same layer cross. The proof is a 2dimensional generalization of a theorem of S. M. Malitz [J. Algorithms 17(1):7184, 1994] on book embeddings. We obtain a Las Vegas algorithm to compute the drawing in O(m log n log log n) time (with high probability).