1. Comparability Graphs and Permutation Graphs Martin Charles Golumbic

2. Review In Lecture #1, we introduced A variety of Intersection graphs – Intervals, Paths in Trees, Arcs on Circles, etc. Chordal graph property and TRO property Hierarchies of graph families – even within chordal graphs (BD/RBD, Block) Algorithmic questions (coloring, clique, etc.)

3. Reading Chapter 1 of Tolerance Graphs (on webpage) Review of some basic graph algorithmics that everyone probably knows (some stuff on webpage, mostly in CLRS and other general books on algorithms and graphs, and Chapters 1 and 2 of AGT&PG (more background) Comparability Graphs (TRO - transitively orientable graphs) Sections 2.4, 5.1, 5.3, 5.4, 5.7 of AGT&PG Permutation Graphs Sections 7.1, 7.2, 7.4, 7.5 of AGT&PG

4. Transitive Orientation (TRO) of G = (V,E) A directed graph H = (V,F) obtained from the undirected graph G by assigning a direction to each undirected edge (an orientation of G) such that it is transitive : x,y  F and yz  F  xz  F We sometimes denote this as F 2  F. A graph G that has a TRO is called a comparability graph.

5. Gamma Relation (forcing orientations) We define a binary relation  on the (orientations of ) edges of the graph G = (V,E): ab  a’ b’  a = a’ and bb’  E or b = b’ and aa’  E The equivalence classes of the transitive closure  * of  are called implication classes.

6. Implication Classes Edges ab and cd are in the same implication class A if there is a  -chain joining them: ab = a 0 b 0  a 1 b 1  a 2 b 2  …  a k b k = cd Note: We are considering sets of directed edges. So, clearly, either A  A -1 =  or A = A -1 where A -1 is the reversal of A.

7. Theorem (Golumbic, 1977) A graph G is a comparability graph if and only if for every implication class A, we have A  A -1 =  This gives an algorithm to recognize comparability graphs, but does not give a TRO. Example. Try it on the triangle K 3

8. The TRO Algorithm Section 5.4 and 5.6 of AGT&PG i := 1 repeat until no edges remain Pick an edge to orient e i Generate its implication class B i Test that B i  B i -1 =  if no, then Fail otherwise, i: = i+1 and remove B i  B i -1 return F = B 1  B 2  …  B k (The TRO)

9. What is the Complexity of the TRO Algorithm?

10. Given a permutation (σ 1,σ 2,σ 3,...) of the numbers 1,2,3,...npermutation A permutation graph has a vertex for each number 1,2,3,...n and an edge between any two numbers that are in reversed order in the permutation, i.e. an edge between any two numbers where the segments cross in the permutation diagram. Permutation Graphs

11. Theorem. A graph G is a permutation graph if and only if both G and its complement G are comparability graphs.

12. Partial Order Dimension To be discussed in a few weeks Section 5.8 and 13.5 of AGT&PG and reference Golumbic, Rotem & Urrutia [1983]

13. Exercises Chapter 1 of Tolerance Graphs (on webpage) Exercises: 1,5,7,11,13,14 Chapter 5 of AGT&PG Exercise 7: Show that if an undirected graph G has no induced subgraph isomorphic to the chordless path P 4 on 4 vertices, the both G and its complement G are comparability graphs. Exercise 11: A binary relation R is called vacuously transitive if R 2 = . Prove that an undirected graph has a vacuously transitive orientation if and only if it is bipartite. Exercise 15: Prove that Algorithm 5.4 correctly computes a maximum weighted clique of a comparability graph. Describe how to implement it to run in O(V+E) time.