Recognition of Linear and Star Variants of Leaf Powers is in P

TL;DR

This paper proves that recognizing linear and star variants of leaf powers, a class of graphs derived from tree structures, can be done efficiently in polynomial time, advancing understanding of their computational complexity.

Contribution

The paper demonstrates that recognizing linear and star variants of leaf powers is solvable in polynomial time, resolving open questions about their computational complexity.

Findings

Recognition algorithms run in polynomial time for both variants.

Linear and star leaf powers are efficiently recognizable.

Addresses open problem from previous research.

Abstract

A $k$-leaf power of a tree $T$ is a graph $G$ whose vertices are the leaves of $T$ and whose edges connect pairs of leaves whose distance in $T$ is at most $k$. A graph is a leaf power if it is a $k$-leaf power for some $k$. Over 20 years ago, Nishimura et al. [J. Algorithms, 2002] asked if recognition of leaf powers was in P. Recently, Lafond [SODA 2022] showed an XP algorithm when parameterized by $k$, while leaving the main question open. In this paper, we explore this question from the perspective of two alternative models of leaf powers, showing that both a linear and a star variant of leaf powers can be recognized in polynomial-time.