独立平行SP-图的螺旋性和直线性平面性检验
Spirality and Rectilinear Planarity Testing of Independent-Parallel SP-Graphs
摘要：
研究了变量嵌入下串-平行图（SP-图）直线平面性的有效检验问题。顶点度最多为3的SP图的线性时间测试算法的设计背后的一个关键因素是，人们可以将注意力限制在每个串联或并联组件的恒定数量的“直线形状”上。为了正式描述这些形状，可以使用螺旋的概念。这一关键因素不再适用于度为4的顶点的SP-图，因为我们证明了其组件的螺旋度的对数下界。即使对于独立平行的SP-图，没有两个平行的组件共享一个极点的界限。尽管如此，通过研究独立平行SP图的螺旋性质，我们能够设计一个线性时间的直线平面性测试算法。
英文摘要：
We study the long-standing open problem of efficiently testing rectilinear planarity of series-parallel graphs (SP-graphs) in the variable embedding setting. A key ingredient behind the design of a linear-time testing algorithm for SP-graphs of vertex-degree at most three is that one can restrict the attention to a constant number of ``rectilinear shapes'' for each series or parallel component. To formally describe these shapes the notion of spirality can be used. This key ingredient no longer holds for SP-graphs with vertices of degree four, as we prove a logarithmic lower bound on the spirality of their components. The bound holds even for the independent-parallel SP-graphs, in which no two parallel components share a pole. Nonetheless, by studying the spirality properties of the independent-parallel SP-graphs, we are able to design a linear-time rectilinear planarity testing algorithm for this graph family.
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