Grounded String Representations of Series-Parallel Graphs without Transitive Edges

2026-03-03

AI Summary

This study addresses the grounded string representation problem for biconnected series-parallel graphs without transitive edges. By integrating structural analysis of series-parallel graphs, orthogonal polyline geometric representations, and linear-time graph traversal techniques, the work establishes—for the first time—that such graphs admit a grounded string representation if and only if they admit a grounded L/inverted-L representation. A linear-time recognition algorithm is provided to decide this property. The main contributions include proving the equivalence of these two representation types within this graph class, and constructing an explicit counterexample that can be represented using L/inverted-L shapes but not with pure L-shapes alone, thereby demonstrating that the expressive power of L/inverted-L representations strictly exceeds that of pure L-representations.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Uncertainty Representations
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• grounded string representation
• series-parallel graphs
• transitive edges
• L-representation
• biconnected graphs

Innovation

• grounded string representation
• series-parallel graphs
• L-reverseL-representation
• transitive edges
• linear-time algorithm