Algorithmic Graph Theory and its Applications

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Algorithmic Graph Theory and its Applications

• Martin Charles Golumbic

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Introduction

• Intersection Graphs
• Interval Graphs
• Greedy Coloring
• The Berge Mystery Story
• Other Structure Families of Graphs
• Graph Sandwich Problems
• Probe Graphs and Tolerance Graphs

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Theconcept of an intersection graph

• applications in computation
• operations research
• molecular biology
• scheduling
• designing circuits
• rich mathematical problems

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Defining some terms

• graph a collection of vertices and edges
• coloring a graph
• assigning a color to every vertex, such
  that
• adjacent vertices have different colors

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• independent set a collection of vertices
• NO two of which are connected
• Example d, e, f or the green set
• clique (or complete set)
• EVERY two of which are connected
• Example a, b, d or c, e

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• complement of a graph
• interchanging the edges and the non-edges

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The complement G
The original graph G
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• directed graph edges have directions
• (possibly both directions)
• orientation exactly ONE direction per edge

cyclic orientation
acyclic orientation
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Interval Graphs

• The intersection graphs of intervals on a line
• - create a vertex for each interval
• - connect vertices when their intervals
  intersect

Task 5
Task 4
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2
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The interval graph G
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History of Interval Graphs

• Hajos 1957 Combinatorics (scheduling)
• Benzer 1959 Biology (genetics)
• Gilmore Hoffman 1964 Characterization
• Booth Lueker 1976 First linear time
  recognition algorithm
• Many other applications
• mobile radio frequency assignment
• VLSI design
• temporal reasoning in AI
• computer storage allocation

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Scheduling Example

• Lectures need to be assigned classrooms at the
  University.
• Lecture a 900-1015
• Lecture b 1000-1200
• etc.
• Conflicting lectures ? Different rooms
• How many rooms?

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Scheduling Example (cont.)
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Scheduling Example (graphs)

1. The interval graph
2. Its complement (disjointness)

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Coloring Interval Graphs

• interval graphs have special properties
• used to color them efficiently
• coloring algorithm sweeps across from left to
  right assigning colors
• in a greedy manner
• This is optimal !

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Coloring Interval Graphs
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Coloring Intervals (greedy)
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Is greedy the best we can do?

• Can we prove optimality?
• Yes It uses the smallest colors.

Proof Let k be the number of colors used.
Look at the point P, when color k was used
first. At P all the colors 1 to k-1 were
busy! We are forced to use k colors at P. And,
they form a clique of size k in the interval
graph.
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Coloring Intervals (greedy)
P (needs 4 colors)
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Coloring Interval Graphs
The clique at point P
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Greedy the best we can do !

• Formally,
• at least k colors are required
• (because of the clique)
• (2) greedy succeeded using k colors.
• Therefore,
• the solution is optimal. Q.E.D.

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Characterizing Interval Graphs

• Properties of interval graphs
• How to recognize them
• Their mathematical structure

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Characterizing Interval Graphs

• Properties of interval graphs
• How to recognize them
• Their mathematical structure

Two properties characterize interval graphs -
The Chordal Graph Property - The co-TRO Property
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The Chordal Graph Property

• chordal graph
• every cycle of length gt 4 has a chord
• (connecting two vertices that are not
  consecutive)
• i.e., they may not contain chordless cycles!

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Interval Graphs are Chordal

• Interval graphs may not contain chordless cycles!
• - i.e., they are chordal. Why?

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Interval Graphs are Chordal

• Interval graphs may not contain chordless cycles!
• - i.e., they are chordal. Why?

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The co-TRO Property

• The transitive orientation (TRO) of the
  complement
• i.e., the complement must have a TRO

Not transitive !
Transitive !
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Interval Graphs are co-TRO

• The complement of an Interval graph has a
  transitive orientation!
• - Why?

The complement is the disjointness graph. So,
orient from the earlier interval to the later
interval.
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Gilmore and Hoffman (1964)

• Theorem
• A graph G is an interval graph
• if and only if G Is chordal and
• its complement G is transitively orientable.

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This provides the basis for the first set of
recognition algorithms in the early 1970s.
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A Mystery in the Library
The Berge Mystery Story

• Six professors had been to the library on the day
  that the rare tractate was stolen.
• Each had entered once, stayed for some time and
  then left.
• If two were in the library at the same time, then
  at least one of them saw the other.
• Detectives questioned the professors and gathered
  the following testimony

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The Facts

• Abe said that he saw Burt and Eddie
• Burt reported that he saw Abe and Ida
• Charlotte claimed to have seen
• Desmond and Ida
• Desmond said that he saw Abe and Ida
• Eddie testified to seeing Burt and Charlotte
• Ida said that she saw Charlotte and Eddie

One of the Professor LIED !! Who was it?
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Solving the Mystery
The Testimony Graph
Clue 1 Double arrows imply TRUTH
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Solving the Mystery
Undirected Testimony Graph
cycle
We know there is a lie, since A, B, I, D is a
chordless 4-cycle.
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Intersecting Intervals cannot form Chordless
Cycles
Burt
Desmond
Abe
No place for Idas interval It must hit both
B and D but cannot hit A. Impossible!
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Solving the Mystery
One professor from the chordless 4-cycle must be
a liar.
There are three chordless 4-cycles A, B, I,
D A, D, I, E A, E, C, D Burt is NOT a
liar He is missing from the second cycle. Ida
is NOT a liar She is missing from the third
cycle. Charlotte is NOT a liar She is missing
from the second. Eddie is NOT a liar He is
missing from the first cycle. WHO IS THE LIAR?
Abe or Desmond ?
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Solving the Mystery (cont.)
WHO IS THE LIAR? Abe or Desmond ?
If Abe were the liar and Desmond truthful, then
A, B, I, D would remain a chordless 4-cycle,
since B and I are truthful. Therefore
Desmond is the liar.
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Was Desmond Stupid or Just Ignorant?

• If Desmond had studied algorithmic graph theory,
  he would have known that his testimony to the
  police would not hold up.

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Many other Families of Intersection Graphs

• Victor Klee, in a paper in 1969
• What are the intersection graphs of arcs in a
  circle?

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Many other Families of Intersection Graphs

• Victor Klee, in a paper in 1969
• What are the intersection graphs of arcs in a
  circle?

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Many other Families of Intersection Graphs

• Victor Klee, in a paper in 1969
• What are the intersection graphs of arcs in a
  circle?
• Klees paper was an implicit challenge
• - consider a whole variety of problems
• - on many kinds of intersection graphs.

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Families of Intersection Graphs

• boxes in the plane
• paths in a tree
• chords of a circle
• spheres in 3-space
• trapezoids, parallelograms, curves of functions
• many other geometrical and topological bodies

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Families of Intersection Graphs

• boxes in the plane
• paths in a tree
• chords of a circle
• spheres in 3-space
• trapezoids, parallelograms, curves of functions
• many other geometrical and topological bodies

• The Algorithmic Problems
• recognize them
• color them
• find maximum cliques
• find maximum independent sets

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A small hierarchy
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The Story Begins
Bell Labs in New Jersey (Spring 1981) John
Klincewicz Suppose you are routing phone calls
in a tree network. Two calls interfere if they
share an edge of the tree. How can you optimally
schedule the calls?
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The Story Begins
Bell Labs in New Jersey (Spring 1981) John
Klincewicz Suppose you are routing phone calls
in a tree network. Two calls interfere if they
share an edge of the tree. How can you optimally
schedule the calls?
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The Story Begins
Bell Labs in New Jersey (Spring 1981) John
Klincewicz Suppose you are routing phone calls
in a tree network. Two calls interfere if they
share an edge of the tree. How can you optimally
schedule the calls?
An Olive Tree Network

• A call is a path between a pair of nodes.
• A typical example of a type of intersection
  graph.
• Intersection here means share an edge.
• Coloring this intersection graph is scheduling
  the calls.

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Edge Intersection Graphs of Paths in a Tree
(EPT graphs)

• tree communication network
• connecting different places
• if two of these paths overlap,
• they conflict and cannot use the
• same resource at the same time.

Two types of intersections share an edge vs
share a node
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EPT graphs
EPT graph share an edge
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VPT graphs
VPT graph share a node
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Some Interesting Theorems

• VPT graphs are chordal
• EPT graphs are NOT chordal

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Some Interesting Theorems

• VPT graphs are chordal
• Buneman, Gavril, Wallace (early 1970's)
• G is the vertex intersection graph of subtrees
  of a tree if and only if it is a chordal graph.
• McMorris Shier (1983)
• A graph G is a vertex intersection graph of
  distinct subtrees of a star if and only if both G
  and its complement are chordal.

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Some Interesting Theorems

• EPT graphs are NOT chordal

An EPT representation of C6 called a 6-pie.
1
6
2
5
3
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Chordless cycles have a unique EPT representation.
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Algorithmic Complexity Results
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Some Interesting Theorems

• Folklore (1970s)
• Every graph G is the edge intersection graph of
  distinct subtrees of a star.

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Degree 3 host trees (continued)
Theorem (1985) All four classes are
equivalent chordal ? EPT ? deg3 EPT ?
VPT ? EPT ? deg3 VPT
What about degree 4?
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Degree 3 host trees (continued)
Theorem (1985) All four classes are
equivalent chordal ? EPT ? deg3 EPT ?
VPT ? EPT ? deg3 VPT
Degree 4 host trees
Theorem (2005) Golumbic, Lipshteyn, Stern
weakly chordal ? EPT ? deg4 EPT
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Weakly Chordal Graphs

• Definition Weakly Chordal Graph
• No induced Cm for m ? 5,
• and
• no induced Cm for m ? 5.

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The Story Continues
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The Interval Graph Sandwich Problem

• Interval problems with missing edges
• Benzers original problem
• partial intersection data
• Is it consistent ?
• Complete data would be recognition interval
  graphs (polynomial)
• Partial data needs a different model and is
  NP-complete

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Interval Graph Sandwich Problem

• given a partially specified graph
• E1 required edges
• E2 optional edges
• E3 forbidden edges
• Can you fill-in some of the optional edges,
• so that the result will be an interval graph?
• Golumbic Shamir (1993) NP-Complete

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Interval Probe Graphs

• A special tractable case of interval sandwich
• Computational biology motivated
• Interval probe graph vertices are partitioned
• P probes N non-probes (independent set)
• can fill-in some of the N x N edges,
• so that the result will be an interval graph
• Motivation

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Example Interval Probe Graphs
Non-Probes are white
Probe graph
NOT a Probe graph no matter how you partition
vertices!
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Tolerance Graphs

• What if you only have 3 classrooms?
• Cancel a Lecture? or show Tolerance?

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Tolerance Graphs
Measured intersection a small, or tolerable
amount of overlap, may be ignored does NOT
produce an edge at least one of them has to be
bothered

• Assignment of positive numbers
• tv (v ?V) such that
• vw ? E if and only if Iv ? Iw ? min
  tv , tw

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Tolerance Graphs Example
c and f will no longer conflict Ic ? If lt
60 min tc , tf
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