Bob and Alice play a game. Bob sends a sequence of positive numbers to Alice and using that information she forms a directed graph.
For each number in the sequence, she splits it into two non-empty parts without any leading zeroes and adds a directed edge from the vertex in the left part to the one in the right. For example, if she gets the integer $12034$, she can add an edge from vertex $1$ to vertex $2034$ or from vertex $120$ to vertex $34$ or from vertex $1203$ to vertex $4$.
12|034is not a valid split because the right part contains a leading zero)
Alice splits each number in such a way that the resultant graph after adding all the edges has no cycles.
Before starting the game, Bob lets Alice know that there is an edge from vertex $1$ to vertex $1010$.
Can Bob find a strategy to send a sequence such that Alice always ends up with a graph containing an edge from vertex $1$ to vertex $21$?