Grounded String Representations of Series-Parallel Graphs without Transitive Edges

Abstract

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.

Figures (26)

• Figure 1: poles not adjacent
• Figure 2: poles adjacent
• Figure 4: curves
• Figure 5: poles not adjacent
• Figure 6: poles adjacent

Theorems & Definitions (23)

• Lemma 1
• proof
• Lemma 1: \ref{['lem:heavyIsCriticalA']}
• theorem 2
• proof : Proof Sketch
• Corollary 2: \ref{['cor:algoA']}
• proof : Proof Sketch
• theorem 3: \ref{['thm:noLA']}
• proof : Proof Sketch
• Lemma 3: \ref{['lem:heavyIsCriticalA']}