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 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 The publication of Berge’s book in the early 1970’s generated a new spurt of interest.

4. 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 The first generation permutation graphs the intersection graphs of permutation diagrams

5. The hierarchy of graph classes Comparability graphs Permutation graphs = Comparability & Co-comparability Interval graphs = Chordal & Co-comparability Chordal graphs Perfect graphs 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: 1.G is a split graph. 2.G and G are chordal graphs. 3.G contains no induced subgraph isomorphic to 2K 2, C 4, or C 5.

7. Recognizing split graphs by their degree sequences Theorem (Hammer and Simeone 1977) Let m = max {i | d i ≥ i  1} Then G is a split graph if and only if Order the vertices by their degree: d 1 ≥ d 2 ≥ … ≥ d n dmdm i Thus, recognizing split graphs is O(n log n).

8. Splittance of a graph 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 Step-by-step transformation of a graph, reducing the stability number at each step. New polynomial time algorithms for several classes of graphs Ebenegger, Hammer and de Werra (1984) 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 Reduction xy 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) 69-88. 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. So, threshold graphs are chordal and co-chordal.

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

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

19. My encounter with threshold graphs New York – Kalamazoo – Keszthey Resource problem: t units available of some commodity agent i requests a i 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 Marty in New York (Columbia, Courant, Bell Labs)

22. 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 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. 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.

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 ( r x, r y ) >  ( t x, t y )

25. Threshold graphs (Chvátal & Hammer 1977) xy ∊ E  ρ ( r x, r y ) >  ( t x, t y )

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

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

29. The parameter space

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