Two Results on Outer-String Graphs

2026-05-12

AI Summary

This study addresses the computational complexity of outer-string graph representations. Specifically, it investigates two problems: first, determining whether a triangle- and pentagon-free graph admits a constrained outer-string representation under a given cyclic vertex ordering; and second, deciding whether a general graph admits an outer-$k$-string representation for any fixed $k \geq 1$. By leveraging graph-theoretic and combinatorial characterizations, the work provides the first forbidden-subgraph characterization for the existence of constrained outer-string representations in graphs excluding $C_3$ and $C_5$, leading to a polynomial-time recognition algorithm. Furthermore, through a reduction-based argument, it establishes that the outer-$k$-string recognition problem is NP-hard for every fixed $k \geq 1$, thereby delineating its computational intractability.

Abstract

An \emph{outer-string representation} of a graph $G$ is an intersection representation of $G$ where vertices are represented by curves (strings) inside the unit disk and each curve has exactly one endpoint on the boundary of the unit disk (the anchor of the curve). Additionally, if each two curves are allowed to cross at most once, we call this an \emph{outer-$1$-string representation} of $G$. If we impose a cyclic ordering on the vertices of $G$ and require the cyclic order of the anchors to respect this cyclic order, such a representation is called a \emph{constrained outer-string representation}. In this paper, we present two results about graphs admitting outer-string representations. Firstly, we show that for a bipartite graph $G$ (and, more generally, for any $\{C_3,C_5\}$-free graph $G$) with a given cyclic order of vertices, we can decide in polynomial time whether $G$ admits a constrained outer-string representation. Our algorithm follows from a characterization by a single forbidden configuration, similar to that of Biedl et al. [GD 2024] for chordal graphs. Secondly, we answer an open question from the same authors and show that determining whether a given graph admits an outer-1-string representation is NP-hard. More generally, we show that it is NP-hard to determine if a given graph $G$ admits an outer-$k$-string representation for any fixed $k\ge1$.

Problem

• outer-string representation
• constrained outer-string representation
• outer-1-string representation
• NP-hard
• C3,C5-free graph

Innovation

• outer-string graph
• constrained representation
• forbidden configuration
• NP-hardness
• polynomial-time algorithm