In Chordal/Interval Vertex Deletion we ask how many vertices one needs to remove from a graph to make it chordal (respectively: interval). We study these problems under the parameterization by treewidth $tw$ of the input graph $G$. On the one hand, we present an algorithm for Chordal Vertex Deletion with running time $2^{O(tw)} \cdot |V(G)|$, improving upon the running time $2^{O(tw^2)} \cdot |V(G)|^{O(1)}$ by Jansen, de Kroon, and Wlodarczyk (STOC'21). When a tree decomposition of width $tw$ is given, then the base of the exponent equals $2^{\omega-1}\cdot 3 + 1$. Our algorithm is based on a novel link between chordal graphs and graphic matroids, which allows us to employ the framework of representative families. On the other hand, we prove that the known $2^{O(tw \log tw)} \cdot |V(G)|$-time algorithm for Interval Vertex Deletion cannot be improved assuming Exponential Time Hypothesis.

翻译：在弦图/区间图顶点删除问题中，我们需要计算从图中移除多少个顶点才能使其成为弦图（或区间图）。本文基于输入图$G$的树宽$tw$参数化研究此类问题。一方面，我们提出一种运行时间为$2^{O(tw)} \cdot |V(G)|$的弦图顶点删除算法，改进了Jansen、de Kroon与Wlodarczyk（STOC'21）提出的$2^{O(tw^2)} \cdot |V(G)|^{O(1)}$算法。当给定宽度为$tw$的树分解时，指数底数等于$2^{\omega-1}\cdot 3 + 1$。该算法基于弦图与图拟阵之间的新颖联系，使我们能够运用代表族框架。另一方面，我们证明已知的运行时间为$2^{O(tw \log tw)} \cdot |V(G)|$的区间图顶点删除算法在指数时间假说下无法被改进。