Inserting an edge into a geometric embedding

• Published in: Computational geometry : theory and applications Vol. 102; p. 101843
• Main Authors: Radermacher, Marcel, Rutter, Ignaz
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2022
• Subjects: Crossings; Disjoint paths; Edge Insertion; Geometric embeddings; Stretchability; Stretchability; Edge Insertion; Geometric embeddings; Crossings; Disjoint paths
• ISSN: 0925-7721
• DOI: 10.1016/j.comgeo.2021.101843

The algorithm to insert an edge e in linear time into a planar graph G with a minimal number of crossings on e[14] is a helpful tool for designing heuristics that minimize edge crossings in topological drawings of general graphs. Unfortunately, not all such topological drawings are stretchable, i.e., there may not exist an equivalent straight-line drawing. That is, there is no planar straight-line drawing Γ of G such that in Γ+e the edge e crosses the same edges as in the topological drawing of G+e and it does so in the same order. This motivates the study of the computational complexity of the problem Geometric Edge Insertion: Given a combinatorially embedded graph G, compute a geometric embedding Γ of G that minimizes the crossings in Γ+e. We give a characterization of the stretchable topological drawings of G+e that also applies to the case where the outer face is fixed; this answers an open question of Eades et al. [8]. Algorithmically, we focus on the case where the outer face is not fixed. We show that Geometric Edge Insertion can be solved efficiently for graphs of maximum degree 5. For the general case, we show a (Δ−2)-approximation, where Δ is the maximum vertex degree of G and an FPT algorithm with respect to the minimum number of crossings. Finally, we consider the problem of testing whether there exists a solution of Geometric Edge Insertion that achieves the lower bound obtained by a topological insertion.