TY - GEN

T1 - Inserting one edge into a simple drawing is hard

AU - Arroyo, Alan

AU - Klute, Fabian

AU - Parada, Irene

AU - Seidel, Raimund

AU - Vogtenhuber, Birgit

AU - Wiedera, Tilo

PY - 2020/1/1

Y1 - 2020/1/1

N2 - A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+ e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ, it can be decided in polynomial time whether there exists a pseudocircle Φ
σ extending σ for which A∪ { Φ
σ} is again an arrangement of pseudocircles.

AB - A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+ e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ, it can be decided in polynomial time whether there exists a pseudocircle Φ
σ extending σ for which A∪ { Φ
σ} is again an arrangement of pseudocircles.

UR - http://www.scopus.com/inward/record.url?scp=85093871748&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-60440-0_26

DO - 10.1007/978-3-030-60440-0_26

M3 - Conference paper

SN - 9783030604394

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 325

EP - 338

BT - Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Revised Selected Papers

A2 - Adler, Isolde

A2 - Müller, Haiko

CY - Leeds, United Kingdom

T2 - 46th International Workshop on Graph-Theoretic Concepts in Computer Science

Y2 - 24 June 2020 through 26 June 2020

ER -