On strongly chordal graphs that are not leaf powers

A common task in phylogenetics is to find an evolutionary tree representing proximity relationships between species. This motivates the notion of leaf powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V and a threshold k such that uv is an edge if and only if the distance between u and v in T is at most k. Characterizing leaf powers is a challenging open problem, along with determining the complexity of their recognition. This is in part due to the fact that few graphs are known to not be leaf powers, as such graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf powers could be characterized by strong chordality and a finite set of forbidden subgraphs. In this paper, we provide a negative answer to this question, by exhibiting an infinite family \G of (minimal) strongly chordal graphs that are not leaf powers. During the process, we establish a connection between leaf powers, alternating cycles and quartet compatibility. We also show that deciding if a chordal graph is \G-free is NP-complete, which may provide insight on the complexity of the leaf power recognition problem

The 4-Steiner Root Problem

International audienceThe 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