Recognizing Geometric Intersection Graphs Stabbed by a Line

TL;DR

This paper proves that recognizing grounded L-graphs and stabbable grid intersection graphs is NP-complete, resolving open problems and advancing understanding of the computational complexity of these geometric intersection graph classes.

Contribution

It establishes NP-completeness for recognizing grounded L-graphs and stabbable grid intersection graphs, answering previously open questions in geometric graph recognition.

Findings

Recognizing grounded L-graphs is NP-complete.

Recognizing stabbable grid intersection graphs is NP-complete.

Both results resolve open problems in geometric graph recognition.

Abstract

In this paper, we determine the computational complexity of recognizing two graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection} graphs. An L-shape is made by joining the bottom end-point of a vertical ($\vert$) segment to the left end-point of a horizontal ($-$) segment. The top end-point of the vertical segment is known as the {\em 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 Jel{\'\i}nek \& T{\"o}pfer (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 $\ell$ stabs a segment $s$, if $s$ intersects $\ell$. 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 \etal (\textsc{O}rder, 2018).