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 a quadratic 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：受二维分层直线图的平面化启发，我们考虑修改图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）询问是否可以通过至多$k$次顶点爆炸操作获得此类图，其中一次顶点爆炸操作将顶点$v$替换为deg$(v)$个度数为1的顶点，每个顶点恰好与原本连接$v$的一条边相连。针对POVE，我们给出运行时间为$O(4^k \cdot m)$的FPT算法和二次核，从而在更一般的设置下改进了Ahmed等人[GD 22]的$O(k^6)$核。类似地，顶点分裂操作将顶点$v$替换为两个不同顶点$v_1$和$v_2$，并将原本连接$v$的边任意分配给$v_1$和$v_2$。类比POVE，我们定义问题变体路径宽度为1的顶点分裂问题（POVS），该问题使用分裂操作而非顶点爆炸。在此我们得到一个线性核和运行时间为$O((6k+12)^k \cdot m)$的算法，这回答了Ahmed等人[GD22]的一个开放问题。最后，我们考虑问题$\Pi$顶点分裂（$\Pi$-VS），它推广了POVS问题，询问是否可以通过至多$k$次顶点分裂操作将给定图转化为特定图类$\Pi$中的图。对于可在一元二阶图逻辑（MSO$_2$）中测试的图类$\Pi$，我们证明$\Pi$-VS问题可表示为MSO$_2$公式，从而在$\Pi$额外具有有界树宽时得到以$k$为参数的$\Pi$-VS的FPT算法。对于使用顶点爆炸操作的问题变体，我们获得相同结果。