  A bicyclic pair is a smooth surface equipped with a pair of smooth divisors
intersecting in two reduced points. Resolutions of self-nodal curves constitute
an important special case. We investigate the logarithmic Gromov-Witten theory
of bicyclic pairs. We establish all-genus correspondences with local
Gromov-Witten theory and open Gromov-Witten theory, and a genus zero
correspondence with orbifold Gromov-Witten theory. For self-nodal curves in
$\mathbb{P}(1,1,r)$ we obtain a closed formula for the genus zero invariants
and relate these to the invariants of local curves. We also establish a
conceptual relationship to the invariants obtained by smoothing the self-nodal
curve. The technical heart of the paper is a qualitatively new analysis of the
degeneration formula for stable logarithmic maps, complemented by torus
localisation and scattering techniques.
