The Contributions of Peter L. Hammer to Algorithmic Graph Theory

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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.  

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Graphs and Hypergraphs
The publication of Berges book in the early
1970s 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

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

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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).
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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.
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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
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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 a(G? ) a(G) ?1
In general, it may grow exponentially
large. But for some graph classes, the
growth can be limited.
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Neighborhood Reduction
x
y
If Nx ? Ny, then delete y. a(G ? y)
a(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.
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Optimal cell flipping to minimize channel
densityin VLSI design and pseudo-Boolean
optimizationEndre Boros, Peter L. Hammer, Michel
Minoux, David J. Rader, Jr.Discrete Applied
Mathematics 90 (1999) 69-88.
Flip selected cells to minimize channel width
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On the complexity of cell flipping in
permutationdiagrams and multiprocessor
scheduling problemsMartin Charles Golumbic, Haim
Kaplan, Elad VerbinDiscrete Mathematics 296
(2005) 25 41
Flip selected cells to minimize channel
thickness i.e., coloring the permutation graph
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Threshold graphs
Probably the most important family of graphs
introduced by Peter Hammer.
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Threshold graphs (Chvátal Hammer 1977)
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Threshold graphs (Chvátal Hammer 1977)
So, threshold graphs are chordal and co-chordal.
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Threshold graphs (Chvátal Hammer 1977)
So, threshold graphs are comparability and
co-comparability.
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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






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My encounter with threshold graphs
New York Kalamazoo Keszthey
Resource problem t units available of some
commodity agent i requests ai units
(i1,,n) all or nothing
A subset S of requests that are satisfiable, form
a stable set
of what kind of graph?
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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,,k1.
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In the 1970s, Peter in Waterloo Marty in New
York (Columbia, Courant, Bell Labs)
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In the 1970s, 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.
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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.
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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 ? (
rank(x), rank(y) ) gt ? ( tolerance(x),
tolerance(y) )
xy ? E ? ? ( rx , ry) gt ? ( tx , ty )
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Threshold graphs (Chvátal Hammer 1977)
xy ? E ? ? ( rx , ry) gt ? ( tx , ty )
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Mix functions and their
rank-tolerance graphs
Remark
Theorem
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Mix functions and their
rank-tolerance graphs
Theorem
is contained in the split graphs.
1. For
2.
3.
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The parameter space
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Conflict and Tolerance in Graph Theory
My next talk Warwick in March 2009
Thank you Peter Thank you RUTCOR