  A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with
leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G
if and only if the distance between x and y in T is at most k. Such a tree T is
called a k-leaf root of G. The computational problem of constructing a k-leaf
root for a given graph G and an integer k, if any, is motivated by the
challenge from computational biology to reconstruct phylogenetic trees. For
fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time.
  In this paper, we propose to study optimal leaf roots of graphs G, that is,
the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G
satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots
is more justified as they yield more probable phylogenetic trees. Lafond's
result does not imply polynomial-time computability of optimal leaf roots,
because, even for optimal k-leaf roots, k may (exponentially) depend on the
size of G. This paper presents a linear-time construction of optimal leaf roots
for chordal cographs (also known as trivially perfect graphs). Additionally, it
highlights the importance of the parity of the parameter k and provides a
deeper insight into the differences between optimal k-leaf roots of even versus
odd k.
  Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect
leaf power, chordal cograph
