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 $k\ge1$.

翻译：一个图$G$的\emph{外弦表示}是$G$的一种交表示，其中顶点由单位圆盘内的曲线（弦）表示，每条曲线恰好有一个端点位于单位圆盘边界上（该曲线的锚点）。此外，如果允许每两条曲线最多相交一次，我们称之为$G$的\emph{外$1$-弦表示}。若对$G$的顶点施加一个循环序，并要求锚点的循环序与该循环序一致，则此类表示称为\emph{约束外弦表示}。本文给出了关于允许外弦表示的图的两个结果。首先，我们证明：对于给定顶点循环序的二部图$G$（更一般地，对于任何不含$C_3$和$C_5$的图$G$），可以在多项式时间内判定$G$是否存在约束外弦表示。该算法基于单一禁止构型的刻画，类似于Biedl等人[GD 2024]对弦图的研究。其次，我们回答了同一作者的公开问题，并证明判定给定图是否存在外$1$-弦表示是NP难的。更一般地，我们证明对于任意固定的$k\ge1$，判定给定图$G$是否存在外$k$-弦表示是NP难的。