The 4-Steiner Root Problem

The k th-power of a graph G is obtained by adding an edge between every two distinct vertices at a distance ≤ k in G. We call G a k-Steiner power if it is an induced subgraph of the k th-power of some tree T. In particular, G is a k-leaf power if all vertices in V (G) are leaf-nodes of T. Our main contribution is a polynomial-time recognition algorithm of 4-Steiner powers, thereby extending the decade-year-old results of (Lin, Kearney and Jiang, ISAAC’00) for k = 1, 2 and (Chang and Ko, WG’07) for k = 3. As a byproduct, we give the first known polynomial-time recognition algorithm for 6-leaf powers. Our work combines several new algorithmic ideas that help us overcome the previous limitations on the usual dynamic programming approach for these problems.