Complexity of finding non-planar rectilinear drawings of graphs

Authors:
Ján Maňuch;Murray Patterson;Sheung-Hung Poon;Chris Thachuk
Affiliations:
Dept. of Computer Science, University of British Columbia, Vancouver BC, Canada and Dept. of Mathematics, Simon Fraser University, Burnaby, BC, Canada;Dept. of Computer Science, University of British Columbia, Vancouver, BC, Canada;Dept. of Computer Science, National Tsing Hua University, Taiwan, R.O.C.;Dept. of Computer Science, University of British Columbia, Vancouver, BC, Canada
Venue:
GD'10 Proceedings of the 18th international conference on Graph drawing
Year:
2010

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Abstract

We study the complexity of the problem of finding non-planar rectilinear drawings of graphs. This problem is known to be NP-complete. We consider natural restrictions of this problem where constraints are placed on the possible orientations of edges. In particular, we show that if each edge has prescribed direction "left", "right", "down" or "up", the problem of finding a rectilinear drawing is polynomial, while finding such a drawing with the minimum area is NP-complete. When assigned directions are "horizontal" or "vertical" or a cyclic order of the edges at each vertex is specified, the problem is NP-complete. We show that these two NP-complete cases are fixed parameter tractable in the number of vertices of degree 3 or 4.