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arXiv · 2407.02412

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

Max Dupré la Tour·Manuel Lafond·Ndiamé Ndiaye·Adrian Vetta
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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 $F_k$.

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Max Dupré la Tour, Manuel Lafond, Ndiamé Ndiaye, Adrian Vetta. 2024-07-02. $k$-Leaf Powers Cannot be Characterized by a Finite Set of Forbidden Induced Subgraphs for $k \geq 5$. https://arxiv.org/abs/2407.02412

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