$k$-Leaf Powers Cannot be Characterized by a Finite Set of Forbidden Induced Subgraphs for $k \geq 5$

TL;DR

This paper proves that for all $k \\geq 5$, $k$-leaf power graphs cannot be characterized by a finite set of forbidden induced subgraphs, unlike the cases for $k \\leq 4$.

Contribution

It establishes that no finite forbidden subgraph characterization exists for $k$-leaf powers when $k \\geq 5$, extending the understanding of their structural complexity.

Findings

Finite forbidden subgraph characterization fails for $k \\geq 5$

Characterization holds only for $k \\leq 4$

Implication for graph class recognition complexity

Abstract

A graph $G=(V,E)$ is a $k$-leaf power if there is a tree $T$ whose leaves are the vertices of $G$ with the property that a pair of leaves $u$ and $v$ induce an edge in $G$ if and only if they are distance at most $k$ apart in $T$. For $k\le 4$, it is known that there exists a finite set $F_k$ of graphs such that the class $L(k)$ of $k$-leaf power graphs is characterized as the set of strongly chordal graphs that do not contain any graph in $F_k$ as an induced subgraph. We prove no such characterization holds for $k\ge 5$. That is, for any $k\ge 5$, there is no finite set $F_k$ of graphs such that $L(k)$ is equivalent to the set of strongly chordal graphs that do not contain as an induced subgraph any graph in $F_k$.