Exact Algorithms for Edge Deletion to Cactus Graphs and Weighted Variants

Abstract

We study exact exponential-time algorithms for Edge Deletion to Cactus. Given a connected graph $G$, the task is to delete a minimum number of edges so that the remaining spanning graph is a connected cactus. Akhtar and Philip (IWOCA 2026) gave an $O^{\ast }(3^n)$-time algorithm for the unweighted problem, where $n$ is the number of vertices in the input graph and the $O^{\ast }(\cdot )$ notation hides polynomial factors. We improve this bound to $O^{\ast }(2^n)$ time and space. More generally, if the deletion costs take at most $q$ distinct nonnegative real values, then the weighted problem can be solved in $O^{\ast }(2^n{n}^{O(q)})$ time and space. Thus every fixed number of distinct costs, and in particular the unweighted case, admits a faster exact algorithm. For nonnegative integer costs of total weight $W$, we obtain an $O^{\ast }(2^n(W+1))$ pseudo-polynomial algorithm, while arbitrary nonnegative real costs admit an $O^{\ast }(3^n)$ exact algorithm.