###### Abstract

In this paper, we determine the computational complexity of recognizing two graph classes, grounded ๐ซ-graphs and stabbable grid intersection graphs. An ๐ซ-shape is made by joining the bottom end-point of a vertical (|) segment to the left end-point of a horizontal (โ) segment. The top end- point of the vertical segment is known as the anchor of the ๐ซ-shape. Grounded ๐ซ-graphs are the intersection graphs of ๐ซ-shapes such that all the ๐ซ-shapesโ anchors lie on the same horizontal line. We show that recognizing grounded ๐ซ-graphs is ๐ญ๐ฏ-complete. This answers an open question asked by Jelรญnek & Tรถpfer (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 ๐ stabs a segment ๐ , if ๐ intersects ๐. A graph ๐บ is a stabbable grid intersection graph (S TAB GIG) if there is a grid intersection representation of ๐บ in which the same line stabs all its segments. We show that recognizing S TAB GIG graphs is ๐ญ๐ฏ-complete, even on a restricted class of graphs. This answers an open question asked by Chaplick et al. (Order, 2018).