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

翻译：对于整数 k ≥ 2，如果存在一棵树 T，其叶集为 V(G)，且对 V(G) 中任意顶点 x, y，边 xy 存在于 G 中当且仅当 x 与 y 在 T 中的距离不超过 k，则称图 G 为 k-叶幂。这样的树 T 被称为 G 的 k-叶根。给定图 G 和整数 k，构造 G 的 k-叶根（若存在）的计算问题，源于计算生物学中重建系统发育树的挑战。对于固定 k，Lafond [SODA 2022] 近期在多项式时间内解决了这一问题。本文提出研究图 G 的最优叶根，即具有最小 k 值的 G 的 k-叶根。因此，G 的所有 k'-叶根均满足 k ≤ k'。从计算生物学角度看，寻求最优叶根更具合理性，因为它们能产生更可能的系统发育树。Lafond 的结果并不意味着最优叶根的多项式时间可计算性，因为即使对于最优 k-叶根，k 可能（指数级）依赖于 G 的规模。本文提出了弦图余图（也称为平凡完美图）的最优叶根的线性时间构造方法。此外，本文强调了参数 k 的奇偶性的重要性，并深入揭示了偶数 k 与奇数 k 的最优 k-叶根之间的差异。关键词：k-叶幂、k-叶根、最优 k-叶根、平凡完美叶幂、弦图余图