$\beta$-Stars or On Extending a Drawing of a Connected Subgraph

TL;DR

This paper studies extending partial graph drawings to full graphs with minimal bends, introducing star complexity, and providing bounds and complexity results for such extensions in planar graphs.

Contribution

It introduces the notion of star complexity for polygons, improves bounds on extension bends, and analyzes the complexity of extending star-shaped faces.

Findings

Bound of min{h/2, β + log2(h) + 1} bends per edge for extension

Extension bounds are tight up to a small additive constant

Complexity of testing star-shaped face extension is analyzed

Abstract

We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing $\Gamma_H$ of an induced connected subgraph $H$ can be extended with at most $\min\{ h/2, \beta + \log_2(h) + 1\}$ bends per edge, where $\beta$ is the largest star complexity of a face of $\Gamma_H$ and $h$ is the size of the largest face of $H$. This result significantly improves the previously known upper bound of $72|V(H)|$ [5] for the case where $H$ is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [13].