A Stick graph G=(A\cup B, E) is the intersection graph of a set A of horizontal segments and a set B of vertical segments in the plane, whose left and respectively bottom endpoints lie on the same ground line with slope -1. These endpoints are respectively called A-origins and B-origins. When a total order is provided for the A-origins, the resulting graphs are called A-Stick graphs. In this paper, we propose a characterization of the class of A-Stick graphs using forced pairs, which are pairs of segments in B with the property that only one left-to-right order of their origins is possible on the ground line. We deduce a recognition algorithm for A-Stick graphs running in O(|A|+|B|+|E|) time, thus improving the running time of O(|A|\cdot |B|) of the best current algorithm. We also introduce the problem of finding, for a Stick graph, a representation using segments of minimum total length. The canonical order on the A- and B-origins, output by our recognition algorithm, allows us to obtain partial results on this problem.
翻译:G=(A\cup B, E)是一组A型水平段和一组B型平面垂直段的交叉图,其左端和底端端端点分别位于与斜度相同的地面线上-1。这些终点分别称为A源点和B源点。当为A源点提供总订单时,由此产生的图表被称为A-Stick图。在本文中,我们建议用强制对对子来描述A-Stick图类,这是B区段的对子,其属性在地面线上只能有一个左对右顺序。我们推导出A-Stick图在O( ⁇ A ⁇ B ⁇ E ⁇ )时间运行的识别算法,从而改进当前最佳算法的O( ⁇ A ⁇ cdot ⁇ B ⁇ )运行时间。我们还提出用最小总长度的分数来查找A-Stick图类的描述问题。A-B源点的直线线线线的属性只有一条。我们通过算法来部分的输出结果。