arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2302.14725cs.DS
收藏

顶点分裂到路径宽至多为1的参数化复杂度

Parameterized Complexity of Vertex Splitting to Pathwidth at most 1

Jakob Baumann, Matthias Pfretzschner, Ignaz Rutter

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文研究通过顶点爆炸或顶点分裂将图修改为路径宽至多1的参数化复杂度，给出改进的FPT算法与更小核，并将结果推广到可用MSO₂检验的图类。

摘要：

受二层直线绘图的平面化问题启发，我们考虑修改一个图，使得所得图的路径宽至多为1。问题“路径宽为1的顶点爆炸”（Pathwidth-One Vertex Explosion，POVE）询问是否可以使用至多$k$次顶点爆炸得到这样的图，其中一次顶点爆炸将顶点$v$替换为deg$(v)$个度为1的顶点，每个顶点恰好与原来关联于$v$的一条边相关联。对于POVE，我们给出了运行时间为$O(4^k \cdot m)$的FPT算法和一个$O(k^2)$核，从而在更一般的设定下改进了Ahmed等人[GD 22]的$O(k^6)$核。类似地，一次顶点分裂将顶点$v$替换为两个不同的顶点$v_1$和$v_2$，并将原来关联于$v$的边任意分配给$v_1$和$v_2$。与POVE类似，我们定义了使用分裂操作代替顶点爆炸的问题变体“路径宽为1的顶点分裂”（Pathwidth-One Vertex Splitting，POVS）。在此我们得到一个线性核和一个运行时间为$O((6k+12)^k \cdot m)$的算法。这回答了Ahmed等人[GD22]提出的一个开放问题。最后，我们考虑问题$Π$顶点分裂（$Π$-VS），它推广了POVS问题，并询问是否可以使用至多$k$次顶点分裂将给定图转化为特定图类$Π$中的图。对于可以在一元二阶图逻辑（MSO$_2$）中检验的图类$Π$，我们证明问题$Π$-VS可以表示为一个MSO$_2$公式，从而在$Π$额外具有有界树宽时，得到以$k$为参数的$Π$-VS的FPT算法。对于使用顶点爆炸的问题变体，我们得到相同的结果。

英文摘要：

Motivated by the planarization of 2-layered straight-line drawings, we consider the problem of modifying a graph such that the resulting graph has pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks whether such a graph can be obtained using at most $k$ vertex explosions, where a vertex explosion replaces a vertex $v$ by deg$(v)$ degree-1 vertices, each incident to exactly one edge that was originally incident to $v$. For POVE, we give an FPT algorithm with running time $O(4^k \cdot m)$ and an $O(k^2)$ kernel, thereby improving over the $O(k^6)$-kernel by Ahmed et al. [GD 22] in a more general setting. Similarly, a vertex split replaces a vertex $v$ by two distinct vertices $v_1$ and $v_2$ and distributes the edges originally incident to $v$ arbitrarily to $v_1$ and $v_2$. Analogously to POVE, we define the problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split operation instead of vertex explosions. Here we obtain a linear kernel and an algorithm with running time $O((6k+12)^k \cdot m)$. This answers an open question by Ahmed et al. [GD22]. Finally, we consider the problem $Π$ Vertex Splitting ($Π$-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class $Π$ using at most $k$ vertex splits. For graph classes $Π$ that can be tested in monadic second-order graph logic (MSO$_2$), we show that the problem $Π$-VS can be expressed as an MSO$_2$ formula, resulting in an FPT algorithm for $Π$-VS parameterized by $k$ if $Π$ additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑