  Barnette's conjecture is an unsolved problem in graph theory. The problem
states that every 3-regular (cubic), 3-connected, planar, bipartite (Barnette)
graph is Hamiltonian. Partial results have been derived with restrictions on
number of vertices, several properties of face-partitions and dual graphs of
Barnette graphs while some studies focus just on structural characterizations
of Barnette graphs. Noting that Spider web graphs are a subclass of Annular
Decomposable Barnette (ADB graphs) graphs and are Hamiltonian, we study ADB
graphs and their annular-connected subclass (ADB-AC graphs). We show that
ADB-AC graphs can be generated from the smallest Barnette graph using recursive
edge operations. We derive several conditions assuring the existence of
Hamiltonian cycles in ADB-AC graphs without imposing restrictions on number of
vertices, face size or any other constraints on the face partitions. We show
that there can be two types of annuli in ADB-AC graphs, ring annuli and block
annuli. Our main result is, ADB-AC graphs having non singular sequences of ring
annuli are Hamiltonian.
