Stick Graphs with 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. [GD'18] considered the recognition problem of stick graphs when no order is given (STICK), when the order of either one of the two sets is given (STICK A ), and when the order of both sets is given (STICK AB ). They showed how to solve STICK AB efficiently. In this paper, we improve the running time of their algorithm, and we solve STICK 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 STICK, STICK A , and STICK AB are all NP-complete. On the positive side, we give an efficient solution for STICK AB with fixed stick lengths if there are no isolated vertices.