Deepening the Relationship between SEFE and C-Planarity

Patrizio Angelini; Giordano Da Lozzo

Home/Archive/arXiv

arXiv · 1404.6175

Deepening the Relationship between SEFE and C-Planarity

Patrizio Angelini·Giordano Da Lozzo

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we deepen the understanding of the connection between two long-standing Graph Drawing open problems, that is, Simultaneous Embedding with Fixed Edges (SEFE) and Clustered Planarity (C-PLANARITY). In his GD'12 paper Marcus Schaefer presented a reduction from C-PLANARITY to SEFE of two planar graphs (SEFE-2). We prove that a reduction exists also in the opposite direction, if we consider instances of SEFE-2 in which the intersection graph is connected. We pose as an open question whether the two problems are polynomial-time equivalent.

Explore related subjects

cs.CC cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Patrizio Angelini, Giordano Da Lozzo. 2014-04-24. Deepening the Relationship between SEFE and C-Planarity. https://arxiv.org/abs/1404.6175

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗