Wikipedia shows graph width $k$ as the degeneracy, an ordering of the vertices $v_1,\ldots , v_k$ for which, if we orient each edge $(v_i, v_j)$ towards $i$ where $i<j$, the maximal degree is at most $k$.
However, I recently came across another definition:
A graph $G = (V, E)$ has width $k$ if there exists a (vertex disjoint-)partition $S = \left\{ S_1, S_2, . . . , S_m \right\}$ of $V$ and a rooted tree $T$ with $S_1, S_2, . . . , S_m$ as its vertices such that $\left|S_i\right| ≤ k$ for each $S_i ∈ S$, and for each $(u, v) ∈ E$ there exists $S_i , S_j ∈ S$ such that $S_j$ is a parent of $S_i$ and $u, v ∈ S_i ∪ S_j$ . The tree $T$ is called the $k$-width decomposition of $G$.
This definition does not seem for me to be equivalent to the classical degeneracy/width definition. It seems quite similar to treewidth, but I'm not sure. I tried proving an equivalence, and failed, and think I might have a counter-example, in which the degeneracy is $2$ and the new width definition is $3$, but not sure I understand the definition exactly, and couldn't find any examples for it online.
So my questions are:
- Are both definitions of width equivalent?
- Any example to feel the new definition of width?
- Is it the same or similar to treewidth?
