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Let M be either a 2-disk or a cylinder, and let f be a smooth function on M that takes constant values on the boundary ∂M, has no critical points on ∂M, and satisfies the condition that near each critical point, f can be expressed locally as a homogeneous polynomial without multiple factors. We define V as either the entire boundary ∂M (for the 2-disk) or one of its boundary components (for the cylinder). Let S′(f,V) denote the group of diffeomorphisms that preserve f and are isotopic to the identity relative to V. We prove that the first Betti number of the f-orbit is equal to the number of orbits generated by the action of S′(f,V) on the internal edges of the Kronrod-Reeb graph associated with f.