Maximum outdegree of internal vertex in a multi-source single sink directed acyclic graph

Question

I'm looking to try and get a bound on the maximum outdegree of an internal vertex in a DAG w/ multiple-source vertexes and a single sink vertex given the following constraints:

1. $\text{deg}^-(\text{sink}) = 2$
2. For any internal vertex, $1 \le \text{deg}^-(v) \le 2$
3. Source vertexes can have any outdegree.

Using the degree sum formula, my gut instinct says that the maximum outdegree of an internal vertex is 2 but I'm having difficulty formally proving it. However, I'm new to graph theory so I'm not sure if I'm even on the right track or if my intuition is leading me astray. Any help would is much appreciated!

Answer 1

Consider the following graph

            -> v_1 ->
            -> v_2 -> v_5 ->
            -> v_3 ->
Source -> v -> v_4 -> v_6 -> sink

i.e. the graph $G=(V,E)$, with $V=\{source,v,v_1,v_2,...,v_6,sink\}$ and $E=\{(source,v),(v,v_1),(v,v_2),(v,v_3),(v,v_4),(v_1,v_5),(v_2,v_5),(v_3,v_6),(v_4,v_6),(v_5,sink),(v_6,sink)\}$. Source and sink here are two vertices and thus sink has in-degree exactly 2.

You can easily add more "layers" in the graph and thus increase $\deg^+(v)$. If you do not bound the total number of vertices in your graph, vertex $v$ can have arbitrarily high out-degree (something around $\deg^+(v)=\frac{n-1}{2}$, where $n=|V|$). So the answer would be that there can only be an upper bound that is depending on the number of vertices. Note however, that this upper bound might even be higher than the one in my example.

(Excuse me for that ugly scetch but I only have my phone with me and I am not allowed to upload pictures.)