Two Results on Outer-String Graphs

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 $k\ge1$.