A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal $1$-planar. For a $3$-connected locally maximal $1$-planar graph $G$, we show the existence of a spanning $3$-connected planar subgraph and prove that $G$ is hamiltonian if $G$ has at most three $3$-vertex-cuts, and that $G$ is traceable if $G$ has at most four $3$-vertex-cuts. Moreover, infinitely many non-traceable $5$-connected $1$-planar graphs are presented.

中文翻译：

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局部极大值 1 平面图的长循环和跨越子图

一个图形是 $1$-planar 如果它在平面上有一个图形，使得每条边最多与另一条边交叉一次。此外，如果此图具有附加属性，即对于两条边的每一次交叉，这些边的末端顶点都会产生一个完整的子图，则该图是局部最大 $1$-平面的。对于 $3$-connected 局部最大 $1$-planar graph $G$，我们证明了一个跨越 $3$-connected 平面子图的存在，并证明 $G$ 是哈密顿分布，如果 $G$ 最多有三个 $3$-顶点-cuts，如果 $G$ 最多有四个 $3$-vertex-cuts，则 $G$ 是可追踪的。此外，无限多个不可追踪的$5$-connected$1$-平面图被呈现。