Answer:
A Maximal Independent Set (MIS) is a set of nodes in a graph such that no two nodes in the set are adjacent. It is called "maximal" because it cannot be further extended by adding more nodes while preserving the independence property.

Size of the smallest MIS in a chain of nine nodes:
To find the size of the smallest MIS in a chain of nine nodes, we need to determine the maximum number of non-adjacent nodes that can be selected without violating the independence property.

A chain of nine nodes can be represented as follows:

1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9

Selection process:
To construct an MIS, we start by selecting a node and then exclude all of its adjacent nodes from the set. We repeat this process until no more nodes can be added to the set.

Let's go step by step to determine the smallest MIS:

1. Select Node 1: We select Node 1 and exclude Node 2 from the set.
MIS: {1}

2. Select Node 3: We select Node 3 and exclude Node 2 and Node 4 from the set.
MIS: {1, 3}

3. Select Node 5: We select Node 5 and exclude Node 4 and Node 6 from the set.
MIS: {1, 3, 5}

4. Select Node 7: We select Node 7 and exclude Node 6 and Node 8 from the set.
MIS: {1, 3, 5, 7}

5. Select Node 9: We select Node 9 and exclude Node 8 from the set.
MIS: {1, 3, 5, 7, 9}

At this point, we cannot select any more nodes as all the nodes have been included in the MIS or are adjacent to the selected nodes. Therefore, the size of the smallest MIS in a chain of nine nodes is 5.

Conclusion:
The size of the smallest MIS in a chain of nine nodes is 5. This means that we can select up to five non-adjacent nodes in the chain while preserving the independence property.