A forbidden induced subgraph characterization of distance-hereditary 5-leaf powers

A graph G is a k -leaf power if there is a tree T such that the vertices of G are the leaves of T and two vertices are adjacent in G if and only if their distance in T is at most k . In this situation T is called a k -leaf root of G . Motivated by the search for underlying phylogenetic trees, the notion of a k -leaf power was introduced and studied by Nishimura, Ragde and Thilikos and subsequently in various other papers. While the structure of 3- and 4-leaf powers is well understood, for k ≥ 5 the characterization of k -leaf powers remains a challenging open problem. present paper, we give a forbidden induced subgraph characterization of distance-hereditary 5-leaf powers. Our result generalizes known characterization results on 3-leaf powers since these are distance-hereditary 5-leaf powers.