AI 中文总结
研究二层k-匹配平面图，证明此类图路径宽度至多为2k + 1并构造达到3⌊k/2⌋ + 1的图，探讨单边识别问题的NP难及固定参数可处理性，还证明双边变体在多项式时间内不能常数因子近似。
AI 中文摘要
如果一个图在平面上的绘制满足对于每条边e，与e交叉的边不包含大小大于k的匹配，则该图为k-匹配平面图。k-匹配平面图类推广了其他超平面图类，如k-平面图和扇形平面图。在二分图的二层绘制中，两个二分划类的顶点放置在两条平行水平线上，边在它们之间绘制为直线段。我们证明每个具有二层k-匹配平面图绘制的图的路径宽度至多为2k + 1。此外，对于每个k≥0，我们构造了一个具有二层k-匹配平面图绘制且路径宽度为3⌊k/2⌋ + 1的图。在算法方面，我们考虑给定一侧顶点固定嵌入的单边识别问题，证明该问题是NP难的，同时证明该问题关于k是固定参数可处理的。最后，我们证明除非P = NP，否则双边变体在多项式时间内不能以任何常数因子近似。
英文摘要
A graph is $k$-matching-planar if it admits a drawing in the plane such that, for every edge $e$, the edges crossing $e$ contain no matching of size greater than $k$. The class of $k$-matching-planar graphs generalizes other beyond-planar graph classes, such as $k$-planar and fan-planar graphs. In a $2$-layer drawing of a bipartite graph, the vertices of the two bipartition classes are placed on two parallel horizontal lines and edges are drawn as straight-line segments between them. We prove that every graph with a $2$-layer $k$-matching-planar drawing has pathwidth at most $2k+1$. Moreover, for every $k \geq 0$, we construct a graph with a $2$-layer $k$-matching-planar drawing whose pathwidth is $3\lfloor k/2\rfloor + 1$. On the algorithmic side, we consider the one-sided recognition problem where a fixed embedding of the vertices on one side is given. We show that this problem is NP-hard. On the other hand, we prove that the problem is fixed-parameter tractable with respect to $k$. Finally, we prove that the two-sided variant cannot be approximated within any constant factor in polynomial time unless $\operatorname{P}=\operatorname{NP}$.
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