Note on the 4- and 5-leaf powers

TL;DR

This paper proves that any 4-leaf power graph with at least one leaf also has a 5-leaf power root, advancing understanding of the relationship between 4- and 5-leaf powers in graph theory.

Contribution

It establishes that 4-leaf power graphs with leaves are also 5-leaf powers, answering a specific open question about leaf power hierarchies.

Findings

Any 4-leaf power with leaves is also a 5-leaf power.

The result applies to graphs with at least one leaf.

Provides insight into the structure of leaf powers.

Abstract

Motivated by the problem of reconstructing evolutionary history, Nishimura et al. defined $k$-leaf powers as the class of graphs $G=(V,E)$ which has a $k$-leaf root $T$, i.e., $T$ is a tree such that the vertices of $G$ are exactly the leaves of $T$ and two vertices in $V$ are adjacent in $G$ if and only if their distance in $T$ is at most $k$. It is known that leaf powers are chordal graphs. Brandst\"adt and Le proved that every $k$-leaf power is a $(k+2)$-leaf power and every 3-leaf power is a $k$-leaf power for $k\geq 3$. They asked whether a $k$-leaf power is also a $(k+1)$-leaf power for any $k\geq 4$. Fellows et al. gave an example of a 4-leaf power which is not a 5-leaf power. It is interesting to find all the graphs which have both 4-leaf roots and 5-leaf roots. In this paper, we prove that, if $G$ is a 4-leaf power with $L(G)\neq \emptyset$, then $G$ is also a 5-leaf power, where $L(G)$ denotes the set of leaves of $G$.