Polynomial-time Recognition of 4-Steiner Powers

The kth-power of a given graph G=(V,E) is obtained from G by adding an edge between every two distinct vertices at a distance at most k in G. We call G a k-Steiner power if it is an induced subgraph of the kth-power of some tree. 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. A graph G is termed k-leaf power if there is some tree T such that: all vertices in V(G) are leaf-nodes of T, and G is an induced subgraph of the kth-power of T. As a byproduct of our main result, we give the first known polynomial-time recognition algorithm for 6-leaf powers