The Contributions of Peter L. The Contributions of Peter L. Hammer to Algorithmic Graph Hammer to Algorithmic Graph

TheoryTheory

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, tocompute the stability number of a graph. In this talk, I will survey some of the fundamental contributions of PeterL. Hammer in graph theory and algorithms, and how they have lead to thedevelopment of new research areas.  

Graphs and HypergraphsGraphs 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

ApplicationsAlgorithmic aspects

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

comparability graphsthose that admit a transitive orientation (TRO)

of its edgeschordal graphs

those that have no chordless cycles ≥ 4interval graphs

the intersection graphs of intervals

on a line

The first generation

permutation graphs

the intersection graphs of

permutation diagrams

The hierarchy of graph The hierarchy of graph classesclasses

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

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 2K2, C4, or C5.

Recognizing split graphs by Recognizing split graphs by their degree sequencestheir degree sequences

Theorem (Hammer and Simeone 1977)

Let m = max {i | di ≥ i 1}Then G is a split graph if and only if

Order the vertices by their degree: d1 ≥ d2 ≥ … ≥ dn

m

i

n

mi

ii dmmd1 1

)1(

dm

i

Thus, recognizing split graphs is O(n log n).

Splittance of a graphSplittance 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

121 )1(

mi

i

mi

i ddmm

One of the few classes where the “editing” problem can be done in polynomial time.

Struction:Struction:Computing the Stability Computing the Stability

NumberNumber

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

An example, An example, fromfrom

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.

Neighborhood ReductionNeighborhood Reduction

x 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.

Optimal cell flipping to minimize channel Optimal cell flipping to minimize channel densitydensityin VLSI design and pseudo-Boolean in VLSI design and pseudo-Boolean optimizationoptimizationEndre Boros, Peter L. Hammer, Michel Minoux, Endre Boros, Peter L. Hammer, Michel Minoux, David J. Rader, Jr.David J. Rader, Jr.Discrete Applied Mathematics Discrete Applied Mathematics 90 (1999) 69-88.90 (1999) 69-88.

Flip selected cells to minimize channel width

On the complexity of cell flipping in On the complexity of cell flipping in permutationpermutationdiagrams and multiprocessor scheduling diagrams and multiprocessor scheduling problemsproblemsMartin Charles Golumbic, Haim Kaplan, Elad Martin Charles Golumbic, Haim Kaplan, Elad VerbinVerbinDiscrete MathematicsDiscrete Mathematics 296 (2005) 25 – 41 296 (2005) 25 – 41

Flip selected cells to minimize channel “thickness” – i.e., coloring the permutation graph

Threshold graphsThreshold graphsProbably the most important family of graphs introduced by Peter Hammer.

Threshold graphs Threshold graphs (Chvátal & (Chvátal & Hammer 1977)Hammer 1977)

Threshold graphsThreshold graphs (Chvátal & (Chvátal & Hammer 1977)Hammer 1977)

So, threshold graphs are chordal and co-chordal.

Threshold graphs Threshold graphs (Chvátal & (Chvátal & Hammer 1977)Hammer 1977)

So, threshold graphs are comparability and co-comparability.

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

My encounter with threshold My encounter with threshold graphsgraphs

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?

Threshold graphs as permutation Threshold graphs as permutation graphsgraphs

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].

In the In the 19701970’s, ’s, Peter in WaterlooPeter in WaterlooMarty in New York Marty in New York (Columbia, (Columbia,

Courant, Bell Labs)Courant, Bell Labs)

In the In the 19701970’s, ’s, Peter in WaterlooPeter in WaterlooMarty in New York Marty in New York (Columbia, (Columbia,

Courant, Bell Labs)Courant, Bell Labs)

In In 19831983, , Peter at RutgersPeter at RutgersMarty in Haifa Marty in Haifa (IBM, Bar-Ilan, U.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.

Peter Hammer as the great Peter Hammer as the great EnablerEnabler

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.

Golumbic and Jamison Golumbic and Jamison 20062006

Rank-Tolerance Graphs Rank-Tolerance GraphsEach 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 )

Threshold graphs Threshold graphs (Chvátal & (Chvátal & Hammer 1977)Hammer 1977)

xy ∊ E ρ ( rx , ry) > ( tx , ty )

Mix functions and their Mix functions and their rank-tolerance graphs rank-tolerance graphs

Remark:

Theorem:

Mix functions and their Mix functions and their rank-tolerance graphs rank-tolerance graphs

Theorem:

is contained in the split graphs.1. For

2.

3.

The parameter spaceThe parameter space

Conflict and Tolerance in Graph Conflict and Tolerance in Graph TheoryTheory

My next talk: Warwick in March 2009: My next talk: Warwick in March 2009:

Thank youPeter

Thank youRUTCOR