A fundamental problem in computational biology is the construction of phylogenetic trees, also called evolutionary trees, for a set of organisms. A graph-theoretic approach takes as input a similarity graph $G$ on the set of organisms, with adjacency denoting evolutionary closeness, and asks for a tree $T$ whose leaves are the set of organisms, with two vertices adjacent in $G$ if and only if the distance between them in the tree is less than some specified distance bound. If this exists $G$ is called a leaf power. Over 20 years ago, [Nishimura et al., J. Algorithms, 2002] posed the question if leaf powers could be recognized in polynomial time. In this paper we explore this still unanswered question from the perspective of two alternative models of leaf powers that have been rather overlooked. These models do not rely on a variable distance bound and are therefore more apt for generalization. Our first result concerns leaf powers with a linear structure and uses a model where the edges of the tree $T$ are weighted by rationals between 0 and 1, and the distance bound is fixed to 1. We show that the graphs having such a model with $T$ a caterpillar are exactly the co-threshold tolerance graphs and can therefore be recognized in $O(n^2)$ time by an algorithm of [Golovach et al., Discret. Appl. Math., 2017]. Our second result concerns leaf powers with a star structure and concerns the geometric NeS model used by [Brandstadt et al., Discret. Math., 2010]. We show that the graphs having a NeS model where the main embedding tree is a star can be recognized in polynomial time. These results pave the way for an attack on the main question, to settle the issue if leaf powers can be recognized in polynomial time.
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