In this paper we investigate some problems related to the Helly properties of circular-arc graphs, which are defined as intersection graphs of arcs of a fixed circle. As such, circular-arc graphs are among the simplest classes of intersection graphs whose models might not satisfy the Helly property. In particular, some cliques of a circular-arc graph might be Helly in some but not all arc intersection models of the graph. Our first result is an alternative proof of a theorem by Lin and Szwarcfiter which asserts that for every circular-arc graph $G$ either every normalized model of $G$ satisfies the Helly property or no normalized model of $G$ satisfies this property. Further, we study the Helly properties of a single clique of a circular-arc graph $G$. We divide the cliques of $G$ into three types: a clique $C$ of $G$ is always-Helly/always-non-Helly/ambiguous if $C$ is Helly in every/no/(some but not all) normalized model of $G$. We provide a combinatorial description for the cliques of each type, and based on it, we devise a polynomial time algorithm which determines the type of a given clique. Finally, we study the Helly Cliques problem, in which we are given an $n$-vertex circular-arc graph $G$ and some of its cliques $C_1, \ldots, C_k$ and we ask if there is an arc intersection model of $G$ in which all the cliques $C_1, \ldots, C_k$ satisfy the Helly property. We show that: (1) the Helly Cliques problem admits a $2^{O(k\log{k})}n^{O(1)}$-time algorithm (that is, it is FPT when parametrized by the number of cliques given in the input), (2) assuming Exponential Time Hypothesis (ETH), the Helly Cliques problem cannot be solved in time $2^{o(k)}n^{O(1)}$, (3) the Helly Cliques problem admits a polynomial kernel of size $O(k^6)$. All our results use a data structure, called a PQM-tree, which maintains all normalized models of a circular-arc graph $G$.

翻译：本文研究了与圆弧图的Helly性质相关的一些问题，圆弧图定义为固定圆上圆弧的交图。因此，圆弧图是最简单的交图类别之一，其模型可能不满足Helly性质。特别地，圆弧图的某些团在某些（但非所有）圆弧交模型中可能具有Helly性质。我们的第一个结果是对Lin与Szwarcfiter定理的另一种证明，该定理断言：对于任意圆弧图$G$，要么$G$的所有归一化模型均满足Helly性质，要么$G$的任何归一化模型均不满足该性质。此外，我们研究了圆弧图$G$中单个团的Helly性质。我们将$G$的团分为三类：团$C$称为始终Helly/始终非Helly/模糊型，若$C$在$G$的每个/无/某些（但非全部）归一化模型中具有Helly性质。我们为每类团提供了组合描述，并基于此设计了多项式时间算法以判定给定团的类型。最后，我们研究了Helly团问题：给定一个$n$顶点圆弧图$G$及其若干团$C_1, \ldots, C_k$，询问是否存在$G$的圆弧交模型，使得所有团$C_1, \ldots, C_k$均满足Helly性质。我们证明：（1）Helly团问题存在$2^{O(k\log{k})}n^{O(1)}$时间算法（即参数化为输入团数量时属于FPT）；（2）假设指数时间假说（ETH），Helly团问题无法在$2^{o(k)}n^{O(1)}$时间内求解；（3）Helly团问题存在大小为$O(k^6)$的多项式核。所有结果均依赖于一种称为PQM树的数据结构，该结构维护了圆弧图$G$的所有归一化模型。