You may want to download the lecture slides that were used for these videos (PDF).
1. Mirsky’s and Dilworth’s Theorem
This video gives a brief review and discussion of two theorems that we will need for this lecture. (3:22)
2. Detecting Cover Graphs
If we are given a graph, is there a way of determining whether it is a cover graph? (5:49)
3. Comparability Graphs
This video defines comparability graphs, and asks whether we can determine whether a given graph is a comparibility graph. (3:47)
4. Detecting Comparability Graphs
Given a graph G, how hard is it to determine whether it is a comparability graph? (9:30)
5. Transitive Orientations
This video offers an alternate definition of a comparability graph, and distinguishes between directed and oriented graphs. (3:34)
6. The P3 Rule and Forbidden Graphs
This video describes the P3, or Vee rule. A transitive graph with an induced P3 must have both edges oriented towards the center vertex, or both edges oriented away from the center vertex. (11:04)
7. The P3 Algorithm
This short video states the algorithm, and gives a brief description of how it works. (1:39)
8. Example
This video explores a second example of the algorithm. (14:58)
9. Another Example
This video explores a third example of the algorithm. (10:15)
10. Gallai’s Theorem
G cannot be a comparibility graph if it contain any of the graphs in a list published by Gallai in 1967. (6:51)
11. Cover Graphs and Orientations
Going back to the idea of a cover graph, we describe another definition of a cover graph that uses directed cycles. Why is it the case that we cannot develop an efficient algorithm to determine whether a graph is a cover graph? (2:22)
12. Interval Orders
This video introduces the idea of an interval order, which is something we will examine in more detail later in this course. (4:01)