Computing Optimal Leaf Roots of Chordal Cographs in Linear Time

Abstract

A graph $G$ is a $k$-leaf power, for an integer $k \geq 2$, if there is a tree $T$ with leaf set $V(G)$ such that, for all distinct 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 $k \leq k'$. 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 $k$.