Vertex Cover Reconfiguration and Beyond

In the Vertex Cover Reconfiguration (VCR) problem, given a graph $G$, 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 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 (regular) graphs of bounded degree, and is solvable in polynomial time on trees and (with some additional restrictions) on cactus graphs.

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