Comparability Graphs and Permutation Graphs

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Comparability Graphs and Permutation Graphs

• Martin Charles Golumbic

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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.)

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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
  AGTPG (more background)
• Comparability Graphs (TRO - transitively
  orientable graphs)
• Sections 2.4, 5.1, 5.3, 5.4, 5.7 of AGTPG
• Permutation Graphs
• Sections 7.1, 7.2, 7.4, 7.5 of AGTPG

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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.
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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.
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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.

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• 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

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The TRO Algorithm

• Section 5.4 and 5.6 of AGTPG
• 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 i1 and remove Bi ? Bi -1
  
• return F B1 ? B2 ? ? Bk (The TRO)

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What is the Complexity of the TRO Algorithm?
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Permutation Graphs
Given a permutation (s1,s2,s3,...) 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.
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Permutation Graphs

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

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Partial Order Dimension

• To be discussed in a few weeks
• Section 5.8 and 13.5 of AGTPG and reference
  Golumbic, Rotem Urrutia 1983

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Exercises

• Chapter 1 of Tolerance Graphs (on webpage)
• Exercises 1,5,7,11,13,14
• Chapter 5 of AGTPG
• 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(VE) time.