Let G be a graph, and $$v \in V(G)$$ and $$S \subseteq V(G)\setminus{v}$$ of size at least k . An important result on graph connectivity due to Perfect states that, if v and S are k -linked, then a ( k −1)-link between a vertex v and S can be extended to a k -link between v and S such that the endvertices of the ( k −1)-link are also the endvertices of the k -link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets S 1 and S 2 are k -linked, then a t -link ( t < k ) between two disjoint sets S 1 and S 2 can be extended to a k -link between S 1 and S 2 such that the endvertices of the t -link are preserved in the k -link. Next, we are able to use these results to show that a 3-connected claw-free graph always has a cycle passing through any given five vertices, but avoiding any other one specified vertex. We also show that this result is sharp by exhibiting an infinite family of 3-connected claw-free graphs in which there is no cycle containing a certain set of six vertices but avoiding a seventh specified vertex. A direct corollary of our main result shows that a 3-connected claw-free graph has a topological wheel minor W k with k ≤ 5 if and only if it has a vertex of degree at least k . Finally, we also show that a graph polyhedrally embedded in a surface always has a cycle passing through any given three vertices, but avoiding any other specified vertex. The result is best possible in the sense that the polyhedral embedding assumption is necessary, and there are infinitely many graphs polyhedrally embedded in surfaces having no cycle containing a certain set of four vertices but avoiding a fifth specified vertex.

中文翻译：

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无爪图和多面体图的循环遍历性

设 G 是一个图，并且 $$v \in V(G)$$ 和 $$S \subseteq V(G)\setminus{v}$$ 的大小至少为 k 。由于 Perfect 的图连通性的一个重要结果指出，如果 v 和 S 是 k 连接的，那么顶点 v 和 S 之间的 ( k -1) 连接可以扩展到 v 和 S 之间的 ak 连接，使得(k -1)-link 的端点也是 k-link 的端点。我们首先证明Perfect结果的推广，证明如果两个不相交的集合S 1 和S 2 是k-连接的，那么两个不相交的集合S 1 和S 2 之间的at-link ( t < k ) 可以扩展为ak S 1 和 S 2 之间的 -link 使得 t -link 的端点保留在 k -link 中。接下来，我们可以使用这些结果来证明一个 3 连通的无爪图总是有一个循环通过任何给定的五个顶点，但避免任何其他指定的顶点。我们还通过展示 3 个连通的无爪图的无限族来表明这个结果是尖锐的，其中没有包含特定的 6 个顶点集但避开第 7 个指定顶点的循环。我们的主要结果的一个直接推论表明，一个 3 连通的无爪图具有一个拓扑轮次 W k 且 k ≤ 5 当且仅当它的顶点度数至少为 k 。最后，我们还表明，多面体嵌入曲面中的图总是有一个循环通过任何给定的三个顶点，但避开任何其他指定的顶点。从多面体嵌入假设是必要的意义上说，结果是最好的，