Grounded String Representations of Series-Parallel Graphs without Transitive Edges

TL;DR

This paper investigates grounded string representations of series-parallel graphs without transitive edges, establishing equivalences between different representation types and providing linear-time testing algorithms.

Contribution

It characterizes when series-parallel graphs admit grounded L-reverseL-representations and grounded string representations, and shows the existence of graphs with specific representation properties.

Findings

Equivalence between grounded L-reverseL and string representations for certain graphs

Linear-time algorithm to test existence of such representations

Existence of graphs with L-reverseL but no L-representation

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.