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

TL;DR

The paper proves that for every $k\ge5$, the class of $k$-leaf power graphs cannot be characterized by a finite set of forbidden induced subgraphs within the strongly chordal graphs. It introduces three gadget graphs (Top, Bot, Interior) to construct an infinite family of minimal non-$k$-leaf powers, $G_{k,n}$, via concatenating Top, $n$ Interior gadgets, and Bot; the main theorem follows by showing that any finite obstruction set would have to contain all $G_{k,n}$, which is impossible. A parallel result extends to linear $k$-leaf powers, requiring a caterpillar-preserving modification of the gadgets. The findings sharpen our understanding of leaf-power structure and imply that a finite obstruction-based recognition framework cannot capture $k$-leaf powers for large $k$, motivating exploration of infinite obstructions or restricted-subclass characterizations with practical recognition implications.

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 $F_k$.

Figures (12)

• Figure 1: The construction of $H_n$
• Figure 2: The $k$-leaf roots of $H_n - \text{Bot}$ and $H_n - \text{Top}$
• Figure 3: The Top Gadget for $k=5$ and $k=6$.
• Figure 4: The $k$-leaf roots of the diamond with $\min_{v\in V(D)\setminus \{b\}} d(b,v)=k-1$. Here the bold edges denote paths of the described length; the dotted edges are the edges of the diamond.
• Figure 5: The interior gadget $I$ for odd $k$. The bold edges signify all possible connections are made.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.0
• Lemma 2.0
• Lemma 2.0
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of \ref{['thm:main']}