The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in R-3 and that any planar graph admits the same even in R-2. For a graph G and d is an element of{2, 3}, let rho(1)(d) (G) denote the minimum number of lines in R-d that together can cover all edges of a drawing of G. For d = 2, G must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results.- For d is an element of {2, 3}, we prove that deciding whether rho(1)(d) ( G) <= k for a given graph G and integer k is there exists R-complete.- Since NP subset of there exists R, deciding rho(1)(d) (G) <= k is NP-hard for d is an element of{2, 3}. On the positive side, we show that the problem is fixed-parameter tractable with respect to k.- Since there exists R subset of PSPACE, both rho(1)(2) (G) and rho(1)(3) (G) are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to rho(1)(2) or rho(1)(3) sometimes require irrational coordinates.- Let rho(2)(3) (G) be the minimum number of planes in R-3 needed to cover a straight-line drawing of a graph G. We prove that deciding whether rho(2)(3) ( G) <= k is NP-hard for any fixed k >= 2. Hence, the problem is not fixed-parameter tractable with respect to k unless P = NP.