Tessellate S.T.E.M.S - Mathematics - College - Set 1 - Problem 2

Alice and Bob are playing a game. $G$ is a connected graph with $n$ vertices, $n \geq 5$ . The players take turn one by one. In each turns, a player either moves his/her position from the vertex he/she is in to any neighboring vertex, or stays in the vertex where he/she was. Their initial positions are given to be distinct vertices. Alice takes the first turn. Bob wins, if at any point of the game, the positions of the two players are the same vertex. Alice wins otherwise. Which of the following conditions are sufficient to ensure a winning strategy for Bob?

This problem is a part of Tessellate S.T.E.M.S.