In a {\em grounded string representation} of a graph there is a horizontal line $\ell$ and each vertex is represented as a simple curve below $\ell$ with one end point on $\ell$ such that two curves intersect if and only if the respective vertices are adjacent. A grounded string representation is a {\em grounded L-reverseL-representation} if each vertex is represented by a 1-bend orthogonal polyline. It is a {\em grounded L-representation} if in addition all curves are L-shaped. We show that every biconnected series-parallel graph without edges between the two vertices of a separation pair (i.e., {\em transitive edges}) admits a grounded L-reverseL-representation if and only if it admits a grounded string representation. Moreover, we can test in linear time whether such a representation exists. We also construct a biconnected series-parallel graph without transitive edges that admits a grounded L-reverseL-representation, but no grounded L-representation.
翻译:在图的一种{\em 接地字符串表示}中,存在一条水平线$\ell$,每个顶点表示为$\ell$下方的一条简单曲线,其一端点位于$\ell$上,且两条曲线相交当且仅当对应的顶点相邻。若每个顶点由一条具有1个折点的正交折线表示,则该接地字符串表示称为{\em 接地L-反L表示}。若进一步所有曲线均为L形,则称为{\em 接地L表示}。我们证明,每个不存在分离对两顶点间边(即{\em 传递边})的双连通串并联图,当且仅当它存在接地字符串表示时,才存在接地L-反L表示。此外,我们可以在线性时间内测试此类表示是否存在。我们还构造了一个无传递边的双连通串并联图,该图存在接地L-反L表示,但不存在接地L表示。