Question 27
Let $G$ be a connected simple graph with vertices $v_1, v_2, \dots, v_{11}$ such that $v_1$ is of degree $2$ and the rest of the vertices are of degree $3$. Assume that for every subset $A$ of vertices with at most $4$ elements, the set of vertices that are not in $A$ but have an edge connecting them to some element of $A$ has at least as many elements as $A$. Find the smallest possible diameter of the graph $G$.
*(Note. A graph is said to be connected if there is a path between every pair of its vertices. The diameter of a graph is the length of the shortest path between the most distanced vertices.)*
*(Note. A graph is said to be connected if there is a path between every pair of its vertices. The diameter of a graph is the length of the shortest path between the most distanced vertices.)*