Common-center graph property (${\mathrm{STAR}}_n(G)$)

All edges of a simple undirected graph have at least one common endpoint. The empty graph is included when there is at least one vertex, and isolated vertices are permitted. This is the star predicate with isolated vertices allowed; a connected star graph is a special case. Its decision-tree depth is the full number $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ of edge bits for $n\ge 3$.

For $n\ge 3$, a full star graph requires every nonincident edge to be certified absent, and fixing those bits suffices. Rejection is certified by at most three present edges with empty common intersection; a triangle needs all three. Hence query certificate complexity is $\max \{3,(\genfrac{}{}{0ex}{}{n-1}{2})\}=\Theta (n^2)$ for this property.

For $n\ge 3$, count the empty graph once, all one-edge graphs once, and each larger accepted graph under its unique center. The latter weighted contribution is $n(n-2)$, giving $1-\left(\genfrac{}{}{0ex}{}{n}{2}\right)+n(n-2)$. Its nonzero value proves evasiveness by the alternating-sum criterion for decision-tree evasiveness.