R. B. Sandeep

Dhanyamol Antony, L. Sunil Chandran, Dalu Jacob et al.

Inversion of a directed graph $D$ with respect to a vertex subset $Y$ is the directed graph obtained from $D$ by reversing the direction of every arc whose endpoints both lie in $Y$. More generally, the inversion of $D$ with respect to a tuple $(Y_1, Y_2, \ldots, Y_\ell)$ of vertex subsets is defined as the directed graph obtained by successively applying inversions with respect to $Y_1, Y_2, \ldots, Y_\ell$. Such a tuple is called a \emph{decycling family} of $D$ if the resulting graph is acyclic. In the \textsc{$k$-Inversion} problem, the input consists of a directed graph $D$ and an integer $k$, and the task is to decide whether $D$ admits a decycling family of size at most $k$. Alon et al.\ (SIAM J.\ Discrete Math., 2024) proved that the problem is NP-complete for every fixed value of $k$, thereby ruling out XP algorithms, and presented a fixed-parameter tractable (FPT) algorithm parameterized by $k$ for tournament inputs. In this paper, we generalize their algorithm to a broader variant of the problem on tournaments and subsequently use this result to obtain an FPT algorithm for \textsc{$k$-Inversion} when the underlying undirected graph of the input is a block graph. Furthermore, we obtain an algorithm for \textsc{$k$-Inversion} on general directed graphs with running time $2^{O(\mathrm{tw}(k + \mathrm{tw}))} \cdot n^{O(1)}$, where $\mathrm{tw}$ denotes the treewidth of the underlying graph.

Marta Piecyk, R. B. Sandeep

Subdividing an edge $uv$ in a graph replaces it by a path $u w v$ with one new vertex. For a graph $H$, the \textsc{$H$-free Subdivision} problem asks whether, given a graph $G$ and an integer $k$, one can destroy all induced copies of $H$ in $G$ by at most $k$ edge subdivisions. We show that the problem is polynomial-time solvable when every component of $H$ is a subdivided star or a subdivided bistar, and at most one component is a subdivided bistar. On the other hand, we prove that \textsc{$H$-free Subdivision} is NP-complete and, assuming the Exponential Time Hypothesis, admits no $2^{o(k)} n^{O(1)}$-time algorithm whenever $H$ satisfies any of the following conditions: \begin{itemize} \item $H$ has minimum degree at least $2$, and the neighborhood of every degree-$2$ vertex induces a $K_2$; \item the vertices of degree at least $3$ in $H$ induce a graph with at least two edges; \item $H$ has a triangle with two vertices of degree at least $3$; \item $H$ contains, as an induced subgraph, the graph obtained from two vertex-disjoint triangles by adding one edge between them; \item $H$ contains exactly one triangle; \item $H$ has girth at least $4$; \item $H$ is a tree with exactly two vertices of degree at least $3$ at distance $2$ or at least $4$. \end{itemize} A simple bounded search-tree algorithm for the problem runs in $2^{O(k)} n^{O(1)}$ time. Thus, for all hardness cases above, this running time is essentially optimal under ETH.