The biclique partition number of a graph. The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.

The tree-depth of a graph. The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as $$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$ Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once). Tree-depth is strictly greater than [[St000536|pathwidth]]. A [[St001120|longest path]] in $G$ has at least $\operatorname{td}(G)$ vertices [3].

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

The dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.

The metric dimension of a graph. This is the length of the shortest vector of vertices, such that every vertex is uniquely determined by the vector of distances from these vertices.

The Colin de Verdière graph invariant.

The book thickness of a graph. The book thickness (or pagenumber, or stacknumber) of a graph is the minimal number of pages required for a book embedding of a graph.

The number of vertices of the largest induced subforest of a graph.

The height of the bicoloured Motzkin path associated with the Dyck path.

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.