Lecture 16 – Interval Order & Interval Graph Algorithms

You may want to download the lecture slides that were used for these videos (PDF).

1. Interval Orders

This video reviews interval orders and describes an example. (1:51)

2. Characterizing Interval Orders

Some posets are interval orders, some are not. Fishburn’s Theorem (1970) gives us a necessary and sufficient condition for determining whether a poset is an interval order. This video provides the proof that an interval order cannot contain a 2 + 2. (7:32)

3. Characterizing Interval Orders, Continued

This video completes our proof by proving sufficiency. (13:55)

4. Interval Order Algorithm

Now that we have a proof of Fishburn’s Theorem, we need an easily implementable algorithm. This video discusses how to find whether a 2 + 2 exists in a poset algorithmically. (6:57)

5. Interval Order Algorithm, Continued

This video discusses how to write a poset as an interval order when a 2 + 2 is not present. (19:07)

6. Interval Order Algorithm (Summary)

This brief video summarizes how the interval order algorithm works. (1:17)

7. More on the Interval Algorithm

This video describes more on the output of the algorithm, and how there are additional exercises in the textbook where you can try to implement the algorithm. (2:30)

8. Recognizing Interval Graphs

If we are given a graph, can we determine whether it is an interval graph? How can we tell? This video gives an implementable and efficient algorithm for answering this question. (4:49)

9. Example of Determining Whether a Graph is an Interval Graph

This video gives an example of the polynomial time algorithm we introduced for determining whether a graph is an interval graph. (15:30)

10. Forbidden Subgraphs for Interval Graphs

This video summarizes the main ideas of this lecture, discusses some of the history behind testing whether a graph is an interval graphs, and lists a number of constructive things that you should be able to do with the theorems and algorithms we introduced in this lecture. (3:03)