Recognizing geometric intersection graphs stabbed by a line

In this paper, we determine the computational complexity of recognizing two graph classes, grounded -graphs and stabbable grid intersection graphs. An -shape is made by joining the bottom end-point of a vertical (|) segment to the left end-point of a horizontal (−) segment. The top end-point of the vertical segment is known as the anchor of the -shape. Grounded -graphs are the intersection graphs of -shapes such that all the -shapes' anchors lie on the same horizontal line. We show that recognizing grounded -graphs is -complete. This answers an open question asked by Jelínek & Töpfer (Electron. J. Comb., 2019).

Grid intersection graphs are the intersection graphs of axis-parallel line segments in which two vertical (similarly, two horizontal) segments cannot intersect. We say that a (not necessarily axis-parallel) straight line ℓ stabs a segment s, if s intersects ℓ. A graph G is a stabbable grid intersection graph (Image 1) if there is a grid intersection representation of G in which the same line stabs all its segments. We show that recognizing Image 1 graphs is -complete, even on a restricted class of graphs. This answers an open question asked by Chaplick et al. (Order, 2018).