### Abstract

This paper introduces and studies the following beyond-planarity problem, which we call h-CLIQUE2PATH PLANARITY. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-CLIQUE2PATH PLANARITY asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, we prove that h-CLIQUE2PATH PLANARITY is NP-complete even when h = 4 and G is a simple 3-plane graph, while it can be solved in linear time when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.

Original language | English (US) |
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Article number | 194 |

Journal | Algorithms |

Volume | 13 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2020 |

### Keywords

- Cliques
- K-planarity
- NP-hardness
- Paths
- Planar graphs
- Polynomial time reduction

### ASJC Scopus subject areas

- Theoretical Computer Science
- Numerical Analysis
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

*Algorithms*,

*13*(8), [194]. https://doi.org/10.3390/A13080194