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 $\Pi$ Vertex Splitting ($\Pi$-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class $\Pi$ using at most $k$ vertex splits. For graph classes $\Pi$ that can be tested in monadic second-order graph logic (MSO$_2$), we show that the problem $\Pi$-VS can be expressed as an MSO$_2$ formula, resulting in an FPT algorithm for $\Pi$-VS parameterized by $k$ if $\Pi$ additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions.

翻译：受双层直线图平面化问题的启发，本文考虑对图进行修改使得结果图的路径宽度至多为1的问题。"路径宽度为1的顶点爆炸问题"(POVE)询问是否可以通过最多$k$次顶点爆炸操作得到这样的图，其中顶点爆炸操作将顶点$v$替换为deg$(v)$个度数为1的顶点，每个新顶点恰好与原图中与$v$相连的一条边相关联。针对POVE问题，我们提出运行时间为$O(4^k \cdot m)$的FPT算法以及$O(k^2)$规模的核化算法，在更一般设条件下改进了Ahmed等人[GD 22]的$O(k^6)$核化结果。类似地，顶点分裂操作将顶点$v$替换为两个不同的顶点$v_1$和$v_2$，并将原与$v$相连的边任意分配给$v_1$和$v_2$。类比POVE，我们定义使用分裂操作替代顶点爆炸的问题变体"路径宽度为1的顶点分裂问题"(POVS)，并给出其线性核化算法以及运行时间为$O((6k+12)^k \cdot m)$的算法，这回答了Ahmed等人[GD22]提出的开放性问题。最后，我们考虑问题$\Pi$顶点分裂($\Pi$-VS)，该问题将POVS推广为询问能否通过最多$k$次顶点分裂操作将给定图转化为属于特定图类$\Pi$的图。对于可在单子二阶图逻辑(MSO$_2$)中验证的图类$\Pi$，我们证明$\Pi$-VS问题可表达为MSO$_2$公式，进而当$\Pi$具有有界树宽时，得到参数化于$k$的$\Pi$-VS问题的FPT算法。对于使用顶点爆炸操作的问题变体，我们获得相同结论。