Every graph with treewidth k admits a tree-decomposition of width <=14k+13 with each vertex in <=deg(v)+1 bags, plus a second decomposition of width <=3k-1 with average three bags per vertex.
Width functions for hypertree decompositions
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Tree decompositions with small width, spread, order and degree
Every graph with treewidth k admits a tree-decomposition of width <=14k+13 with each vertex in <=deg(v)+1 bags, plus a second decomposition of width <=3k-1 with average three bags per vertex.