1. The Contributions of Peter L. Hammer to Algorithmic Graph Theory
Martin Charles Golumbic (University of Haifa)
Abstract
Peter L. Hammer authored or co-authored more than 240 research papers
during his professional career. Of these, about 20% are in graph theory
-- alone about equal to the whole career of most people!
Together with colleagues, his work includes introducing the families of
threshold graphs and split graphs, graph parameters such as the Dilworth
number and the splittance of a graph, and the operation called struction, to
compute the stability number of a graph.
In this talk, I will survey some of the fundamental contributions of Peter
L. Hammer in graph theory and algorithms, and how they have lead to the
development of new research areas.  

3. Graphs and Hypergraphs
The publication of Berge’s book in the early 1970’s generated a new spurt of interest.
Basic structured families of graphs
comparability graphs and chordal graphs
interval graphs and permutation graphs
other classes of intersection graphs and of perfect graphs
Applications
Algorithmic aspects

4. those that admit a transitive orientation (TRO) of its edges
The first generation
comparability graphs
those that admit a transitive orientation (TRO)
of its edges
chordal graphs
those that have no chordless cycles ≥ 4
interval graphs
the intersection graphs of intervals on a line
permutation graphs
the intersection graphs of permutation diagrams

5. The hierarchy of graph classes
Perfect graphs
Comparability graphs
Chordal graphs
Interval graphs =
Chordal &
Co-comparability
Permutation graphs =
Comparability &
Co-comparability
What ????? =
Chordal &
Co-chordal
The answer was provided by Földes and Hammer (1977): Split graphs

6. A graph G is a split graph if its vertices can be partitioned into an independent set and a clique.
Theorem (Földes and Hammer 1977)
The following are equivalent:
G is a split graph.
G and G are chordal graphs.
G contains no induced subgraph isomorphic to
2K2, C4, or C5.

7. Recognizing split graphs by their degree sequences
Order the vertices by their degree: d1 ≥ d2 ≥ … ≥ dn
Theorem (Hammer and Simeone 1977)
Let m = max {i | di ≥ i  1}
Then G is a split graph if and only if dm i
Thus, recognizing split graphs is O(n log n).

8. Splittance of a graph Theorem (Hammer and Simeone 1977)
Definition: the minimum number of edges to be added or erased in order to make G into a split graph.
Theorem (Hammer and Simeone 1977)
The splittance depends only on the degree sequence, and equals
One of the few classes where the “editing” problem can be done in polynomial time.

9. Struction: Computing the Stability Number
Ebenegger, Hammer and de Werra (1984)
Step-by-step transformation of a graph,
reducing the stability number at each step.
New polynomial time algorithms for several classes of graphs
CN-free graphs,
CAN-free,
and others

10. An example, from Struction Revisited,
Alexe, Hammer, Lozin & de Werra (2004)
Choose a pivot x in G.
Replace x and its neighbors with
some new vertices and edges.
Obtain G such that α(G ) = α(G) 1
In general, it may grow
exponentially large.
But for some graph classes,
the growth can be limited.

11. Neighborhood Reductionx y
If N[x]  N[y], then delete y.
α(G  {y}) = α(G)
i.e., no change in stability number
Theorem (Golumbic and Hammer 1988)
Neighborhood reduction can be applied to a circular-arc graph to bring it to a canonical form. The stability number can then be easily calculated.

12. Optimal cell flipping to minimize channel density in VLSI design and pseudo-Boolean optimization Endre Boros, Peter L. Hammer, Michel Minoux, David J. Rader, Jr. Discrete Applied Mathematics 90 (1999)
Flip selected cells to minimize channel width

13. On the complexity of cell flipping in permutation diagrams and multiprocessor scheduling problems Martin Charles Golumbic, Haim Kaplan, Elad Verbin Discrete Mathematics 296 (2005) 25 – 41
Flip selected cells to minimize channel “thickness” – i.e., coloring the permutation graph

14. Threshold graphs
Probably the most important family of graphs introduced by Peter Hammer.

15. Threshold graphs (Chvátal & Hammer 1977)

16. Threshold graphs (Chvátal & Hammer 1977)
So, threshold graphs are chordal and co-chordal.

17. Threshold graphs (Chvátal & Hammer 1977)
So, threshold graphs are comparability and co-comparability.

18. Mahadev and Peled, Threshold Graphs and Related Topics, 1995
Berge,
Graphs and Hypergraphs, 1970
Golumbic,
Algorithmic Graph Theory and Perfect Graphs, 1980
Mahadev and Peled, Threshold Graphs and Related Topics, 1995
Threshold Graphs
Perfect Graphs

19. My encounter with threshold graphs
New York – Kalamazoo – Keszthey
Resource problem: t units available of some commodity
agent i requests ai units (i=1,…,n) [all or nothing]
A subset S of requests that are satisfiable, form a stable set…
… of what kind of graph?

20. Threshold graphs as permutation graphs
Theorem (Golumbic, 1976)
A graph G is a threshold graph if and only if
G is the permutation graph of a “shuffle product” of [1,2,3,…,k] [n,n-1,…,k+1].

21. In the 1970’s,. Peter in Waterloo
In the 1970’s, Peter in Waterloo Marty in New York (Columbia, Courant, Bell Labs)

22. In 1983, Peter at Rutgers Marty in Haifa (IBM, Bar-Ilan, U.Haifa)
In the 1970’s, Peter in Waterloo Marty in New York (Columbia, Courant, Bell Labs)
In 1983, Peter at Rutgers Marty in Haifa (IBM, Bar-Ilan, U.Haifa)
Peter gave me my “first break” into the journal editorial world,
first as a Guest Editor for a special issue of DM, then as an Editorial Board member of the new DAM.

23. Peter Hammer as the great Enabler
Bringing many, many visitors to RUTCOR.
Welcoming collaborative environment.
Encouraging new talent around the world.
Supporting seasoned talent.
Hundreds of new ideas were born at RUTCOR.
Ron Shamir and I introduced the Graph Sandwich Problem while both visiting Rutgers.
Peter gave me a “second big break”:
He enabled me to become the Founder and Editor-in-Chief of the Annals of Mathematics and Artificial Intelligence.

24. Golumbic and Jamison 2006 Rank-Tolerance Graphs
Each vertex receives
A rank indicating its tendency for having edges (conflict)
A tolerance indicating its tendency for not having edges
such that (x,y) ∊ E(G) if and only if
ρ ( rank(x), rank(y) ) >  ( tolerance(x), tolerance(y) )
xy ∊ E  ρ ( rx , ry) >  ( tx , ty )

25. Threshold graphs (Chvátal & Hammer 1977)
xy ∊ E  ρ ( rx , ry) >  ( tx , ty )

27. Mix functions and their rank-tolerance graphs
Remark:
Theorem:

28. Mix functions and their rank-tolerance graphs
Theorem:
1. For
is contained in the split graphs. 2. 3.

29. The parameter space

30. Conflict and Tolerance in Graph Theory
My next talk: Warwick in March 2009:
Conflict and Tolerance in Graph Theory
Thank you
Peter
RUTCOR