The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar connected graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be \NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions.

翻译：给定图的部分绘制，在满足表示约束的条件下寻找其扩展的任务在文献中得到了广泛研究，已有经典结果为平面拓扑绘制及超平面拓扑绘制等基本表示提供了高效算法。本文针对平面连通图的弯折最小正交绘制这一最基本几何图绘制表示的扩展问题展开研究。虽然该问题已知为NP困难问题，但考虑仅需绘制图中小部分子图的情形是自然的研究方向。我们在此证明，当以缺失子图规模为参数时，该问题具有固定参数可解性。算法基于多重创新要素，融合几何与组合论证：包括识别用于平面顶点放置的弯折等价区域的新型图表示、建立该辅助图的树宽上界，以及设计全局点网格以将弯折与顶点的可能放置离散化为各区域局部有界子网格。