Recognizing Geometric Intersection Graphs Stabbed by a Line
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In this paper, we determine the computational complexity of recognizing two graph classes, grounded L-graphs and stabbable grid intersection graphs. An L-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 L-shape. Grounded L-graphs are the intersection graphs of L-shapes such that all the L-shapes' anchors lie on the same horizontal line. We show that recognizing grounded L-graphs is NP-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 (StabGIG) if there is a grid intersection representation of G in which the same line stabs all its segments. We show that recognizing StabGIG graphs is NP-complete, even on a restricted class of graphs. This answers an open question asked by Chaplick (Order, 2018).
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