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:
- Start with an edge $(v_1, v_2)$, a $2$-tree.
- 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 $n$.
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?