发表机构
Department of Mathematics, Mar Ivanios College, University of Kerala; Department of Mathematics, Sree Narayana College; Department of Mathematics, Mahatma Gandhi College; Instituto de Informática, Universidade Federal de Goiás（喀拉拉大学马伊瓦尼奥斯学院数学系; 斯里纳拉亚纳学院数学系; 甘地学院数学系; 戈亚斯联邦大学信息学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究循环凸性下图的 Helly 数、Radon 数和秩的计算复杂性与极值结构，证明相关阈值决策问题在平面图中仍为 NP-难和 W[1]-难，并刻画参数取 n-1 和 n-2 的图类。
AI 中文摘要
本文研究了循环凸性下图的三个基本凸性参数，即 Helly 数、Radon 数和秩。我们首先研究这些参数的计算复杂性。对于每个参数，我们考虑相关的阈值决策问题，即确定给定图的参数是否至少为某个指定整数。我们证明这三个问题在参数化为阈值时都是 NP-难的和 W[1]-难的。此外，我们通过证明即使输入限制为最大度至多 6 的平面图，NP-难性仍然成立，从而强化了这些结果。我们还关注对应于这些参数极值的连通图的结构性质。特别地，我们刻画了三个参数达到值 n-1 和 n-2 的图类，其中 n 是 G 的阶。
英文摘要
In this paper, we investigate three fundamental convexity parameters of graphs under cycle convexity, namely the Helly number, Radon number, and rank. We first study the computational complexity of these parameters. For each of these parameters, we consider the associated threshold decision problem of determining whether the parameter of a given graph is at least a prescribed integer. We establish that all three problems are $\NP$-hard and $\W[1]$-hard when parameterized by the threshold. Moreover, we strengthen these results by showing that the $\NP$-hardness persists even when the input is restricted to planar graphs of maximum degree at most $6$. We also focus on the structural properties of connected graphs corresponding to extremal values of these parameters. In particular, we characterize the graph classes for which the three parameters attain the values $n-1$ and $n-2$, where $n$ is the order of $G$.
