Lecture 15 – Cover Graphs, Comparability Graphs, & Transitive Orientations

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 Pmust 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)