Recognizing Stick Graphs with and without Length Constraints

Abstract

Stick graphs are intersection graphs of horizontal and vertical line segments that all touch a line of slope $-1$ and lie above this line. De Luca et al. [De Luca et al. GD'18] considered the recognition problem of stick graphs when no order is given ($\textsf{STICK}$), when the order of either one of the two sets is given ($\textsf{STICK}_{\textsf A}$), and when the order of both sets is given ($\textsf{STICK}_{\textsf{AB}}$). They showed how to solve $\textsf{STICK}_{\textsf{AB}}$ efficiently. In this paper, we improve the running time of their algorithm, and we solve $\textsf{STICK}_{\textsf A}$ efficiently. Further, we consider variants of these problems where the lengths of the sticks are given as input. We show that these variants of $\textsf{STICK}$, $\textsf{STICK}_{\textsf A}$, and $\textsf{STICK}_{\textsf{AB}}$ are all NP-complete. On the positive side, we give an efficient solution for $\textsf{STICK}_{\textsf{AB}}$ with fixed stick lengths if there are no isolated vertices.

Planar Straight-line Realizations of 2-Trees with Prescribed Edge Lengths

“…The problem of producing drawings of graphs with geometric constraints is a core topic for Graph Drawing [3,4,5,11,14,23,24,27,34,43,45]. In this context, a classic question is the one of testing if a planar graph can be drawn planarly and straight-line with prescribed edge lengths.…”

On the complexity of recognizing Stick graphs

“…For the former variant, a characterization theorem and a quadratic algorithm based on it have been proposed in [8]. A different approach, using a line that sweeps the ordered origins from left to right along the ground line and progressively computes the solutions has been proposed in [7], and yielded a linear algorithm for the former variant, as well as a quadratic algorithm for the latter variant. A third approach [17] extended the ideas in [8] to the more general case where only the origins in A are ordered, and proposed a left-to-right search that is able to build in linear time the digraph recording all the solutions (i.e., the orderings of the origins in A ∪ B for which a Stick representation exists).…”

“…Another variant of Stick graphs has been proposed in [7]. In this case, the length of the segment representing each vertex is provided.…”

“…Recognizing Stick graphs with fixed length segments is NP-complete in the three cases: when no order is provided on the set of origins, when only the origins in A (equivalently, in B) are ordered, and when the origins in both A and B are ordered. Further subvariants where isolated vertices are not admitted are also NP-complete, except in the third case, where this subvariant is polynomially solvable [7].…”

Stick graphs: examples and counter-examples

“…STICK is an open problem. Variants of STICK, assuming that the order of the A-origins and/or the order of the B-origins are known, have been considered and solved in [13,6], including in the case where the lengths of the segments are known. However, very little is known about Stick graphs in general.…”