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

Palindromic complexity of trees

We consider finite trees with edges labeled by letters on a finite alphabet $\varSigma$. Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid $\varSigma^*$. The set of all such words defines the language of the tree. In this paper, we investigate the palindromic complexity of trees and provide hints for an upper bound on the number of distinct palindromes in the language of a tree.Comment: Submitted to the conference DLT201