It is well known that any graph admits a crossing-free straight-line drawing
in R3 and that any planar graph admits the same even in
R2. For a graph G and d∈{2,3}, let ρd1(G) denote
the minimum number of lines in Rd 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∈{2,3}, we prove that deciding whether ρd1(G)≤k for a
given graph G and integer k is ∃R-complete.
- Since NP⊆∃R, deciding ρd1(G)≤k is NP-hard for d∈{2,3}. On the positive side, we show that the problem
is fixed-parameter tractable with respect to k.
- Since ∃R⊆PSPACE, both ρ21(G) and
ρ31(G) are computable in polynomial space. On the negative side, we show
that drawings that are optimal with respect to ρ21 or ρ31
sometimes require irrational coordinates.
- Let ρ32(G) be the minimum number of planes in R3 needed
to cover a straight-line drawing of a graph G. We prove that deciding whether
ρ32(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