In this work, we study the parameter hull number in a recently defined graph convexity called Cycle Convexity, whose definition is motivated by related notions in Knot Theory.

For a graph $G=(V,E)$, define the interval function in the Cycle Convexity as ${\mathrm{I}}_{\mathrm{cc}}(S)=S\cup \{v\in V(G)|\text{there is a cycle }C\text{ in }G\text{ such that }V(C)\setminus S=\{v\}\}$, for every $S\subseteq V(G)$. We say that $S\subseteq V(G)$ is convex if ${\mathrm{I}}_{\mathrm{cc}}(S)=S$. The convex hull of $S\subseteq V(G)$, denoted by $\mathrm{Hull}(S)$, is the inclusion-wise minimal convex set $S^{\prime }$ such that $S\subseteq S^{\prime }$. A set $S\subseteq V(G)$ is called a hull set if $\mathrm{Hull}(S)=V(G)$. The hull number of G in the cycle convexity, denoted by ${\mathrm{hn}}_{\mathrm{cc}}(G)$, is the cardinality of a smallest hull set of G.

We first focus on the class of planar graphs, as the main motivation for the definition of ${\mathrm{hn}}_{\mathrm{cc}}(G)$ stems from Knot Theory and occurs when G is a 4-regular planar graph. We prove that: the hull number of a 4-regular planar graph is at most half of its number of vertices and that such bound is tight; and that deciding whether ${\mathrm{hn}}_{\mathrm{cc}}(G)\le k$, provided a positive integer k and a planar graph G, is an $\mathsf{\text{NP}}$-complete problem.

On the positive side, we present polynomial-time algorithms to compute the hull number in the cycle convexity of chordal graphs, $P_4$-sparse graphs, and grids.

在这项工作中，我们研究了最近定义的称为循环凸性的图凸性中的参数外壳数，其定义是由结理论中的相关概念激发的。

对于图表$G=（V,乙）$，将循环凸性中的区间函数定义为${\mathrm{我}}_{\mathrm{抄送}}（S）=S\cup \{v\epsilon V（G）|\text{有一个循环 }C\text{ 在 }G\text{ 这样 }V（C）\setminus S=\{v\}\}$，对于每一个$S\subseteq V（G）$。我们这么说$S\subseteq V（G）$是凸的，如果${\mathrm{我}}_{\mathrm{抄送}}（S）=S$。的凸包为$S\subseteq V（G）$，表示为$\mathrm{船体}（S）$, 是包含式最小凸集$S^{\prime }$这样$S\subseteq S^{\prime }$。一套$S\subseteq V（G）$称为船体集，如果$\mathrm{船体}（S）=V（G）$。G在循环凸性中的外壳数，表示为${\mathrm{恩}}_{\mathrm{抄送}}（G）$，是G的最小外壳集的基数。

我们首先关注平面图的类别，作为定义的主要动机${\mathrm{恩}}_{\mathrm{抄送}}（G）$源于纽结理论，当G是 4-正则平面图时发生。我们证明：4-正则平面图的外壳数最多为其顶点数的一半，并且该界限是紧界；并决定是否${\mathrm{恩}}_{\mathrm{抄送}}（G）\le k$，假设有一个正整数k和一个平面图G，则为$\mathsf{\text{NP}}$-完整的问题。

从积极的一面来看，我们提出了多项式时间算法来计算弦图的循环凸度中的船体数，${磷}_4$-稀疏图表和网格。