Characterizing graphs of small carving-width

We characterize all graphs that have carving-width at most  kk for k=1,2,3k=1,2,3. In particular, we show that a graph has carving-width at most 3 if and only if it has maximum degree at most 3 and treewidth at most 2. This enables us to identify the immersion obstruction set for graphs of carving-width at most 3.