To prove this, let's break down the problem and use some basic graph theory principles.

Definitions and Problem Breakdown

1. Acyclic Graph: This is a graph with no cycles, essentially a forest (a collection of trees).
2. Leaves: Leaves are vertices of degree 1, meaning each leaf has only one edge connected to it.
3. Connected Components: These are maximal subgraphs in which any two vertices are connected by paths.

The goal is to prove that an acyclic graph with 100 leaves and no isolated nodes has strictly less than 50 connected components if and only if it has at least one vertex of degree 3 or more.

Step-by-Step Proof

1. Assume the Graph Has No Vertex of Degree 3 or More:

   • If no vertex has degree 3 or more, then each vertex has a degree of either 1 or 2.
   • However, if every vertex has degree 2, we would have a cycle, which contradicts the fact that the graph is acyclic.
   • Therefore, each connected component in this case must be a "star" or a path. A star with 2 leaves is just an edge, while a star with exactly one internal vertex and two leaves has an internal vertex of degree 2.
   • With 100 leaves and no higher degree vertices, each component would need at least two leaves (for each path or small star), resulting in at least 50 components.

   Conclusion: If no vertex has degree 3 or more, we cannot have fewer than 50 connected components.

2. Assume There Is At Least One Vertex of Degree 3 or More:

   • If there exists a vertex with degree 3 or more, we can have a larger tree structure where multiple leaves are connected to a single internal vertex.
   • This configuration allows a single connected component to contain multiple leaves.
   • For example, one tree could consist of one internal vertex connected to multiple leaves, which would decrease the overall number of connected components needed to reach 100 leaves.
   • Thus, having vertices of degree 3 or more allows us to "merge" multiple leaves into fewer connected components, resulting in fewer than 50 components.

   Conclusion: If there is a vertex of degree 3 or more, it’s possible to have fewer than 50 connected components.

Final Statement

Therefore, we have shown that:

• If the graph has no vertex of degree 3 or more, it must have at least 50 connected components.
• If the graph has at least one vertex of degree 3 or more, it can have fewer than 50 connected components.

This completes the proof of the "if and only if" condition.

Would you like more details on any specific part, or do you have questions?

---

Here are 5 related questions for further exploration:

1. What is the minimum number of vertices needed to construct a tree with a given number of leaves and a specified number of internal vertices?
2. How does this proof change if we allow isolated nodes in the graph?
3. What if the condition was "at least 60 connected components"? How would the proof change?
4. Can this concept be extended to graphs with cycles (i.e., to non-acyclic graphs)?
5. What types of acyclic graphs (other than trees) can meet specific leaf and degree requirements?

Tip: In any graph problem, breaking down definitions and checking base cases often leads to clearer insights.