The minimal number of edges to add or remove to make a graph a cograph. A cograph is a graph that can be obtained from the one vertex graph by complementation and disjoint union.

The independence gap of a graph. This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph. In particular, this statistic is $0$ for well covered graphs

The maximin edge-connectivity for choosing a subgraph. This is $\max_X \min(\lambda(G[X]), \lambda(G[V\setminus X]))$, where $X$ ranges over all subsets of the vertex set $V$ and $\lambda$ is the edge-connectivity of a graph.

The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.

The lettericity of a graph. Let $D$ be a digraph on $k$ vertices, possibly with loops and let $w$ be a word of length $n$ whose letters are vertices of $D$. The letter graph corresponding to $D$ and $w$ is the graph with vertex set $\{1,\dots,n\}$ whose edges are the pairs $(i,j)$ with $i < j$ sucht that $(w_i, w_j)$ is a (directed) edge of $D$.

The degree of the minimal polynomial of the largest eigenvalue of a graph.

The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$. In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$. We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.

The minimal number of edges to add or remove to make a graph a line graph.

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.