What is the correct definition of $k$-tree?

Question

As the title says, what is the correct definition of $k$-tree? There are several papers that talk about $k$-trees and partial $k$-trees as alternative definitions for graphs with bounded treewidth, and I've seen many seemingly incorrect definitions. For example, at least one place defines $k$-trees as follows:

A graph is called a $k$-tree if and only if either $G$ is the complete graph with $k$ vertices, or $G$ has a vertex $v$ with degree $k − 1$ such that $G \setminus v$ is a $k$-tree. A partial $k$-tree is any subgraph of a $k$-tree.

According to this definition, one can create the following graph:

1. Start with an edge $(v_1, v_2)$, a $2$-tree.
2. For $i=1\ldots n$, create a vertex $v_i$ and make it adjacent to $v_{i-1}$ and $v_{i-2}$.

Doing this would create a strip of $n$ squares with diagonals. Similarly, we can start creating a band from the first square in a direction orthogonal to the strip above. Then, we would have the first row and first column of an $n \times n$ grid. Filling in the grid is easy by creating vertices and joining them to the vertices to its above and to its left.

The end result is a graph that contains an $n\times n$ grid, which, in effect, is known to be of treewidth $n$.

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A correct definition of $k$-trees has to be the following:

A graph is called a $k$-tree if and only if either $G$ is a complete graph with $k$ vertices, or $G$ has a vertex $v$ with degree $k-1$ such that the neighbor of $v$ forms a $k$-clique, and $G \setminus v$ is a $k$-tree.

Then, the grid-like graph described as above cannot be created.

Am I correct?

Answer 1

I basically agree with you, with just a tiny modification:

A graph $G$ is a $k$-tree if and only if either $G$ is a complete graph with $k+1$ vertices, or $G$ has a vertex $v$ such that the (open) neighborhood of $v$ forms a $k$-clique, and $G - v$ is a $k$-tree.

In other words, $v$ should have degree $k$, instead $k-1$ in your definition.

I personally prefer the bottom-up definition, but this is just a matter of taste:

• The complete graph on $k+1$ vertices is a $k$-tree.
• A $k$-tree $G$ with $n+1$ vertices ($n\ge k+1$) can be constructed from a $k$-tree $H$ with $n$ vertices by adding a vertex adjacent to exactly $k$ vertices that form a $k$-clique in $H$.
• No other graphs are $k$-trees.

This definition is a slightly modified version of the definition from Pinar Heggernes' lecture notes.