  We show that an edge-dominating cycle in a $2K_2$-free graph can be found in
polynomial time; this implies that every 1/(k-1)-tough $2K_2$-free graph admits
a k-walk, and it can be found in polynomial time. For this class of graphs,
this proves a long-standing conjecture due to Jackson and Wormald (1990).
Furthermore, we prove that for any \epsilon>0 every (1+\epsilon)-tough
$2K_2$-free graph is prism-Hamiltonian and give an effective construction of a
Hamiltonian cycle in the corresponding prism, along with few other similar
results.
