Vertex Cover Reconfiguration and Beyond

In the Vertex Cover Reconfiguration (VCR) problem, given graph $G = (V, E)$, positive integers $k$ and $\ell$, and two vertex covers $S$ and $T$ of $G$ of size at most $k$, we determine whether $S$ can be transformed into $T$ by a sequence of at most $\ell$ vertex additions or removals such that every operation results in a vertex cover of size at most $k$. Motivated by recent results establishing the W[1]-hardness of VCR when parameterized by $\ell$, we delineate the complexity of the problem restricted to various graph classes. In particular, we show that VCR remains W[1]-hard on bipartite graphs, is NP-hard but fixed-parameter tractable on graphs of bounded degree, and is solvable in time polynomial in $|V(G)|$ on cactus graphs. We prove W[1]-hardness and fixed-parameter tractability via two new problems of independent interest.