Recognizing 2-Layer and Outer $k$-Planar Graphs
Yasuaki Kobayashi, Yuto Okada, Alexander Wolff
TL;DR
The paper investigates the recognition problem for two locally constrained graph classes—2-layer $k$-planar and outer $k$-planar graphs—under fixed-parameter settings. It introduces degree-reduction rules and a dynamic-programming approach to obtain XP algorithms for both problems, and refines one result to an $FPT$ algorithm when the order on one layer is specified, with running time $2^{O(k\log k)}n^{O(1)}$. It also establishes hardness barriers: XNLP-hardness for the general recognition problems and weak NP-hardness for the weighted one-sided variant, highlighting intrinsic complexity barriers even for small $k$ or restricted inputs. These results delineate the tractability landscape for recognizing constrained planar-like graphs and identify precise conditions under which efficient algorithms are feasible. The findings have implications for graph drawing applications and related parameterized complexity inquiries in beyond-planarity regimes.
Abstract
The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural parameter. Two well-known variants of the problem are 2-layer crossing minimization and circular crossing minimization, where every vertex must lie on one of two layers, namely two parallel lines, or a circle, respectively. Both variants are NP-hard, but FPT with respect to the natural parameter. Recently, a local version of the crossing number has also received considerable attention. A graph is $k$-planar if it admits a drawing with at most $k$ crossings per edge. In contrast to the crossing number, recognizing $k$-planar graphs is NP-hard even if $k=1$. In this paper, we consider the two above variants in the local setting. The $k$-planar graphs that admit a straight-line drawing with vertices on two layers or on a circle are called 2-layer $k$-planar and outer $k$-planar graphs, respectively. We study the parameterized complexity of the two recognition problems with respect to $k$. For $k=0$, both problems can easily be solved in linear time. Two groups independently showed that outer 1-planar graphs can also be recognized in linear time [Hong et al., Algorithmica 2015; Auer et al., Algorithmica 2016]. One group asked whether outer 2-planar graphs can be recognized in polynomial time. Our main contribution consists of XP-algorithms for recognizing 2-layer $k$-planar graphs and outer $k$-planar graphs. We complement these results by showing that both recognition problems are XNLP-hard. This implies that both problems are W$[t]$-hard for every $t$ and that it is unlikely that they admit FPT-algorithms. On the other hand, we present an FPT-algorithm for recognizing 2-layer $k$-planar graphs where the order of the vertices on one layer is specified.