In Dual Marching Cubes the dual of an octree is tessellated via the standard marching cubes method. But the classical marching cubes algorithm operates only on cubes with eight vertices (it uses signs at eight corners of the cube to construct a bitmask and select the proper configuration of triangles), while cells of octree duals can actually have more than eight edges.
For example, in the following picture a quadtree dual is shown, where the lower left cell has 8 links to neighboring dual vertices:
although the marching squares algorithm needs only 4 corner vertices (4-bit masks - 16 cases) for contouring the grid.
And how to build the surface in this case? Which edges to select to form a bitmask used in the marching cubes/squares algorithm as the algorithm needs only 8/4 points as input?