10 research outputs found
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