1. Algorithmic Graph Theory and its Applications
Martin Charles Golumbic
Algorithmic Graph Theory

2. Algorithmic Graph Theory
Introduction
Intersection Graphs
Interval Graphs
Greedy Coloring
The Berge Mystery Story
Other Structure Families of Graphs
Graph Sandwich Problems
Probe Graphs and Tolerance Graphs
Algorithmic Graph Theory

3. The concept of an intersection graph
applications in computation
operations research
molecular biology
scheduling
designing circuits
rich mathematical problems
Algorithmic Graph Theory

4. Algorithmic Graph Theory
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
Algorithmic Graph Theory

5. Algorithmic Graph Theory
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 }
Algorithmic Graph Theory

6. Algorithmic Graph Theory
complement of a graph:
interchanging the edges and the non-edges __
The complement G
The original graph G
Algorithmic Graph Theory

7. Algorithmic Graph Theory
directed graph: edges have directions
(possibly both directions)
orientation: exactly ONE direction per edge
cyclic orientation
acyclic orientation
Algorithmic Graph Theory

8. Interval Graphs The intersection graphs of intervals on a line:
- create a vertex for each interval
- connect vertices when their intervals intersect
Jan
Feb
Mar
Apr
May
Jun
July
Sep
Oct
Nov
Dec
Phase 1
Phase 2
Phase 3
Task 5
Task 4 1 2 3
The interval graph G 4 5

9. 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
Algorithmic Graph Theory

10. Scheduling Example
Lectures need to be assigned classrooms at the University.
Lecture #a: 9:00-10:15
Lecture #b: 10:00-12:00
etc.
Conflicting lectures  Different rooms
How many rooms?

11. Scheduling Example (cont.)

12. Scheduling Example (graphs)
The interval graph
Its complement (disjointness)

13. 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 !
Algorithmic Graph Theory

14. Coloring Interval Graphs
Algorithmic Graph Theory

15. Coloring Intervals (greedy)
Algorithmic Graph Theory

16. 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.
Algorithmic Graph Theory

17. Coloring Intervals (greedy)
P (needs 4 colors)
Algorithmic Graph Theory

18. Coloring Interval Graphs
The clique at point P
Algorithmic Graph Theory

19. 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.
Algorithmic Graph Theory

20. Characterizing Interval Graphs
Properties of interval graphs
How to recognize them
Their mathematical structure
Algorithmic Graph Theory

21. 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
Algorithmic Graph Theory

22. The Chordal Graph Property
every cycle of length > 4 has a chord
(connecting two vertices that are not consecutive)
i.e., they may not contain chordless cycles!
Algorithmic Graph Theory

23. Interval Graphs are Chordal
Interval graphs may not contain chordless cycles!
- i.e., they are chordal. Why?
Algorithmic Graph Theory

24. Interval Graphs are Chordal
Interval graphs may not contain chordless cycles!
- i.e., they are chordal. Why?
Algorithmic Graph Theory

25. Algorithmic Graph Theory
The co-TRO Property
The transitive orientation (TRO) of the complement
i.e., the complement must have a TRO
Not transitive !
Transitive !
Algorithmic Graph Theory

26. 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.
Algorithmic Graph Theory

27. Algorithmic Graph Theory
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. __
This provides the basis for the first set of recognition algorithms in the early 1970’s.
Algorithmic Graph Theory

28. 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:

29. One of the Professor LIED !! Who was it?
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?

30. Solving the Mystery The Testimony Graph Clue #1:
Double arrows imply TRUTH

31. Solving the Mystery Undirected Testimony Graph
cycle
We know there is a lie, since {A, B, I, D} is a chordless 4-cycle.

32. Intersecting Intervals cannot form Chordless Cycles
Burt
Desmond
Abe
No place for Ida’s interval:
It must hit both B and D but cannot hit A.
Impossible!

33. 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 ?

34. 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.

35. 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.
Algorithmic Graph Theory

36. Many other Families of Intersection Graphs
Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs in a circle?’’
Algorithmic Graph Theory

37. Many other Families of Intersection Graphs
Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs in a circle?“
Algorithmic Graph Theory

38. Many other Families of Intersection Graphs
Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs in a circle?“
Klee’s paper was an implicit challenge
- consider a whole variety of problems
- on many kinds of intersection graphs.
Algorithmic Graph Theory

39. 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
Algorithmic Graph Theory

40. 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
Algorithmic Graph Theory

41. Algorithmic Graph Theory
A small hierarchy
Algorithmic Graph Theory

42. Algorithmic Graph Theory
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?
Algorithmic Graph Theory

43. Algorithmic Graph Theory
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?
Algorithmic Graph Theory

44. Algorithmic Graph Theory
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.
Algorithmic Graph Theory

45. 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
Algorithmic Graph Theory

46. Algorithmic Graph Theory
EPT graphs
EPT graph
share an edge
Algorithmic Graph Theory

47. Algorithmic Graph Theory
VPT graphs
VPT graph
share a node
Algorithmic Graph Theory

48. Some Interesting Theorems
VPT graphs are chordal
EPT graphs are NOT chordal
Algorithmic Graph Theory

49. 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.
Algorithmic Graph Theory

50. Some Interesting Theorems
EPT graphs are NOT chordal
An EPT representation of C6
called a “6-pie”. 1 6 2 5 3 4
Chordless cycles have a unique EPT representation.
Algorithmic Graph Theory

51. Algorithmic Complexity Results
Algorithmic Graph Theory

52. Some Interesting Theorems
Folklore (1970’s)
Every graph G is the edge intersection graph of distinct subtrees of a star.
Algorithmic Graph Theory

53. Degree 3 host trees (continued)
Theorem (1985): All four classes are equivalent:
chordal  EPT  deg3 EPT
 VPT  EPT  deg3 VPT
What about degree 4?
Algorithmic Graph Theory

54. 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
Algorithmic Graph Theory

55. Algorithmic Graph Theory
Weakly Chordal Graphs
Definition Weakly Chordal Graph
No induced Cm for m  5,
and
no induced Cm for m  5.
Algorithmic Graph Theory

56. Algorithmic Graph Theory
The Story Continues
Algorithmic Graph Theory

57. The Interval Graph Sandwich Problem
Interval problems with missing edges
Benzer’s 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
Algorithmic Graph Theory

58. 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
Algorithmic Graph Theory

59. Algorithmic Graph Theory
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
Algorithmic Graph Theory

60. Example: Interval Probe Graphs
Non-Probes are white
Probe graph
NOT a Probe graph no matter how you partition vertices!
Algorithmic Graph Theory

61. Algorithmic Graph Theory
Tolerance Graphs
What if you only have 3 classrooms?
Cancel a Lecture? or show Tolerance?
Algorithmic Graph Theory

62. Algorithmic Graph Theory
Tolerance Graphs
Measured intersection:
small, or ``tolerable’’ amount of overlap, may be ignored does NOT produce an edge
at least one of them has to be ``bothered’’
Algorithmic Graph Theory

63. Algorithmic Graph Theory
Tolerance Graphs
Measured intersection:
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}
Algorithmic Graph Theory

64. Tolerance Graphs: Example
c and f will no longer conflict
| Ic  If | < 60 = min {tc , tf}
Algorithmic Graph Theory

65. More on Algorithmic Graph Theory