An \emph{orientation} of an undirected graph $G=(V,E)$ is a directed graph $G'=(V,E')$ that has the same set $V$ of vertices and whose set $E'$ of directed edges is obtained by choosing a direction on each of the edges of $G$. That is, for each edge $u\edg v$ in $E$ we put in $E'$ either the directed edge $u\dedg v$ or the directed edge $v\dedg u$, but \emph{not both}.

Let $G=(V,E)$ be an undirected graph with 2 or more nodes and let $a,b$ be two distinct nodes in $V$. Prove that $G$ has some orientation that is a DAG in which $a$ is a source and $b$ is a sink.

To prove that any undirected graph $ G = (V, E) $ with $ |V| \geq 2 $ has an orientation $ G' = (V, E') $ such that $ G' $ is a Directed Acyclic Graph (DAG), $ a $ is a source, and $ b $ is a sink, we proceed as follows:

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### **Step 1: Orientation of Edges**
An **orientation** of $ G $ is a directed graph $ G' = (V, E') $ where each edge $ u \edg v \in E $ is replaced with exactly one of $ u \dedg v $ or $ v \dedg u $, but not both. Our goal is to ensure this orientation results in a DAG where $ a $ is a source (no incoming edges) and $ b $ is a sink (no outgoing edges).

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### **Step 2: Topological Sorting**
1. If $ G $ is connected, we can view $ G $ as a single connected component.
   - Choose a spanning tree $ T $ of $ G $ rooted at $ a $. The spanning tree ensures there are no cycles among its edges.
   - Direct all edges in $ T $ away from $ a $. This ensures $ a $ has no incoming edges.
2. If $ G $ is not connected, repeat this process in the connected component of $ a $. For other connected components, assign arbitrary orientations.

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### **Step 3: Adding Edges to Avoid Cycles**
1. For edges $ e = u \edg v $ not in the spanning tree $ T $, direct them to ensure no cycles are created. Specifically:
   - If $ u $ appears before $ v $ in the topological order of $ T $, direct $ u \dedg v $.
   - Otherwise, direct $ v \dedg u $.
2. This ensures $ G' $ is acyclic because all orientations preserve the topological order of vertices.

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### **Step 4: Ensuring $ b $ is a Sink**
1. In the final orientation $ G' $, ensure $ b $ is a sink by redirecting any outgoing edges from $ b $ to point toward $ b $.
   - Since $ b $ has no outgoing edges, this does not create cycles or violate the acyclic property.

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### **Step 5: Verification**
1. $ a $ is a source because all edges incident to $ a $ are directed away from it in the spanning tree $ T $.
2. $ b $ is a sink because all edges incident to $ b $ are directed toward it.
3. $ G' $ is a DAG because the topological order constructed from the spanning tree is preserved, and no cycles are introduced.

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### **Conclusion**
We have constructed an orientation $ G' = (V, E') $ of $ G $ such that:
1. $ G' $ is a DAG.
2. $ a $ is a source.
3. $ b $ is a sink.

Thus, such an orientation always exists.

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Would you like a detailed example illustrating this process? Here are related questions to explore further:

1. What is the significance of a spanning tree in constructing a DAG?
2. How can we efficiently determine a topological order for a directed graph?
3. What happens if $ G $ is not connected? How does the proof adjust?
4. Can we generalize this result to weighted graphs?
5. How does the choice of root node $ a $ affect the orientation of the graph?

**Tip**: When working with graphs, try visualizing orientations using small examples to verify acyclic properties and node-specific constraints.

Learn how to prove that any undirected graph with 2 or more nodes can be oriented as a Directed Acyclic Graph (DAG) such that one specified node is a source and another is a sink. This proof involves concepts from graph theory, including spanning trees and graph orientation.

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