Comparability Graphs and
Permutation Graphs
Martin Charles Golumbic
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.)
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
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
F2  F .
A graph G that has a TRO is called a comparability graph.
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.
Implication Classes
Edges ab and cd are in the same implication class A
if there is a -chain joining them:
ab = a0b0  a1b1  a2b2  …  akbk = 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.
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 K3
The TRO Algorithm
Section 5.4 and 5.6 of AGT&PG
i := 1
repeat until no edges remain
Pick an edge to orient ei
Generate its implication class Bi
Test that Bi  Bi -1 = 
if no, then Fail
otherwise, i: = i+1 and remove Bi  Bi -1
return F = B1  B2  …  Bk
(The TRO)
What is the Complexity of the TRO Algorithm?
Permutation Graphs
Given a permutation (σ1,σ2,σ3,...) of the numbers 1,2,3,...n
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
• Theorem. A graph G is a permutation graph if
and only if both G and its complement G are
comparability graphs.
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]
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 P4 on 4
vertices, the both G and its complement G are comparability
graphs.
Exercise 11: A binary relation R is called vacuously
transitive if R2 = . 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.