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 \(\textbf{tw}\) of the input graph G. On the one hand, we present an algorithm for Chordal Vertex Deletion with running time \(2^{\mathcal {O}(\textbf{tw})} \cdot |V(G)|\) , improving upon the running time \(2^{\mathcal {O}(\textbf{tw}^2)} \cdot |V(G)|^{\mathcal {O}(1)}\) by Jansen, de Kroon, and Włodarczyk (STOC’21). When a tree decomposition of width \(\textbf{tw}\) 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 Interval Vertex Deletion cannot be solved in time \(2^{o(\textbf{tw}\log \textbf{tw})} \cdot |V(G)|^{\mathcal {O}(1)}\) assuming the Exponential Time Hypothesis.
