Between 2- and 3-colorability

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Between 2- and 3-colorability
Rutgers University
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The problem
O
Independent Set
G
X
Bipartite Graph
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The problem
O
Independent Set
G
X
Bipartite Graph

• tree

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The problem
O
Independent Set
G
X
Bipartite Graph

• tree

• forest

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The problem
O
Independent Set
G
X
Bipartite Graph

• tree

• forest

• of bounded degree

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The problem
O
Independent Set
G
X
Bipartite Graph

• tree

• forest

• of bounded degree

• complete bipartite

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Examples

• Trees NP-complete
• A. Brandstädt, V. B. Le, T. Szymczak, The
  complexity of some problems related to graph
  3-colorability. Discrete Appl. Math. 89 (1998)
  59--73.

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Examples

• Trees NP-complete
• A. Brandstädt, V. B. Le, T. Szymczak, The
  complexity of some problems related to graph
  3-colorability. Discrete Appl. Math. 89 (1998)
  59--73.

• Forest NP-complete

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Examples

• Trees NP-complete
• A. Brandstädt, V. B. Le, T. Szymczak, The
  complexity of some problems related to graph
  3-colorability. Discrete Appl. Math. 89 (1998)
  59--73.

• Forest NP-complete

• Graphs of bounded vertex degree NP-complete
• J. Kratochvíl, I. Schiermeyer, On the
  computational complexity of (O, P)-partition
  problems. Discuss. Math. Graph Theory 17 (1997)
  253--258.

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Examples

• Trees NP-complete
• A. Brandstädt, V. B. Le, T. Szymczak, The
  complexity of some problems related to graph
  3-colorability. Discrete Appl. Math. 89 (1998)
  59--73.

• Forest NP-complete

• Graphs of bounded vertex degree NP-complete
• J. Kratochvíl, I. Schiermeyer, On the
  computational complexity of (O, P)-partition
  problems. Discuss. Math. Graph Theory 17 (1997)
  253--258.

• Complete bipartite Polynomial
• A. Brandstädt, P.L. Hammer, V.B. Le, V. Lozin,
  Bisplit graphs. Discrete Math. 299 (2005) 11--32.

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Question
Is there any boundary separating difficult
instances of the (O,P)-partition problem from
polynomially solvable ones?
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Question
Is there any boundary separating difficult
instances of the (O,P)-partition problem from
polynomially solvable ones?
Yes ?
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Hereditary classes of graphs

• Definition.
• A class of graphs P is hereditary if X?P
  implies X-v?P for any vertex v?V(X)

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Hereditary classes of graphs

• Definition.
• A class of graphs P is hereditary if X?P
  implies X-v?P for any vertex v?V(X)

Examples perfect graphs (bipartite, interval,
permutation graphs), planar graphs, line graphs,
graphs of bounded vertex degree.
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Speed of hereditary properties
E.R. Scheinerman, J. Zito, On the size of
hereditary classes of graphs. J. Combin. Theory
Ser. B 61 (1994) 16--39.
Alekseev, V. E. On lower layers of a lattice of
hereditary classes of graphs. (Russian) Diskretn.
Anal. Issled. Oper. Ser. 1 4 (1997) 3--12.
J. Balogh, B. Bllobás, D. Weinreich, The speed of
hereditary properties of graphs. J. Combin.
Theory Ser. B 79 (2000) 131--156.
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Lower Layers

• constant
• polynomial
• exponential
• factorial

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Lower Layers

• constant
• polynomial
• exponential
• factorial

• planar graphs

• permutation graphs

• line graphs

• graphs of bounded vertex degree

• graphs of bounded tree-width

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Minimal Factorial Classes of graphs

• Bipartite graphs
• 3 subclasses
• Complements of bipartite graphs
• 3 subclasses
• Split graphs, i.e., graphs partitionable into an
  independent set and a clique
• 3 subclasses

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Three minimal factorial classes of bipartite
graphs

• P1 The class of graphs of vertex degree at
  most 1

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Three minimal factorial classes of bipartite
graphs

• P1 The class of graphs of vertex degree at
  most 1

• P2 Bipartite complements to graphs in P1

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Three minimal factorial classes of bipartite
graphs

• P1 The class of graphs of vertex degree at
  most 1

• P2 Bipartite complements to graphs in P1

• P3 2K2-free bipartite graphs (chain or
  difference graphs)

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(O,P)-partition problem
Let P be a hereditary class of bipartite graphs
Problem. Determine whether a graph G admits a
partition into an independent set and a graph in
the class P
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(O,P)-partition problem
Let P be a hereditary class of bipartite graphs
Problem. Determine whether a graph G admits a
partition into an independent set and a graph in
the class P
Conjecture
If P contains one of the three minimal factorial
classes of bipartite graphs, then the
(O,P)-partition problem is NP-complete. Otherwise
it is solvable in polynomial time.
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Polynomial-time results
Theorem. If P contains none of the three minimal
factorial classes of bipartite graphs, then the
(O,P)-partition problem can be solved in
polynomial time.
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Polynomial-time results
Theorem. If P contains none of the three minimal
factorial classes of bipartite graphs, then the
(O,P)-partition problem can be solved in
polynomial time.
If P contains none of the three minimal factorial
classes of bipartite graphs, then P belongs to
one of the lower layers

• exponential
• polynomial
• constant

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Exponential classes of bipartite graphs
Theorem. For each exponential class of bipartite
graphs P, there is a constant k such that for any
graph G in P there is a partition of V(G) into at
most k independent sets such that every pair of
sets induces either a complete bipartite or an
empty (edgeless) graph.
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Exponential classes of bipartite graphs
Theorem. For each exponential class of bipartite
graphs P, there is a constant k such that for any
graph G in P there is a partition of V(G) into at
most k independent sets such that every pair of
sets induces either a complete bipartite or an
empty (edgeless) graph.
(O,P)-partition
2-sat
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NP-complete results
J. Kratochvíl, I. Schiermeyer, On the
computational complexity of (O,P)-partition
problems. Discuss. Math. Graph Theory 17 (1997)
253--258.
Theorem. If P is a monotone class of graphs
different from the class of empty (edgeless)
graphs, then the (O,P)-partition problem is
NP-complete.
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NP-complete results
J. Kratochvíl, I. Schiermeyer, On the
computational complexity of (O,P)-partition
problems. Discuss. Math. Graph Theory 17 (1997)
253--258.
Theorem. If P is a monotone class of graphs
different from the class of empty (edgeless)
graphs, then the (O,P)-partition problem is
NP-complete.
Corollary. The (O,P)-partition problem is
NP-complete if P is the class of graphs of vertex
degree at most 1.
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One more result
Yannakakis, M. Node-deletion problems on
bipartite graphs. SIAM J. Comput. 10 (1981), no.
2, 310--327.
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One more result
Yannakakis, M. Node-deletion problems on
bipartite graphs. SIAM J. Comput. 10 (1981), no.
2, 310--327.
Let P be a hereditary class of bipartite graphs
Problem(P). Given a bipartite graph G, find a
maximum induced subgraph of G belonging to P.
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One more result
Yannakakis, M. Node-deletion problems on
bipartite graphs. SIAM J. Comput. 10 (1981), no.
2, 310--327.
Let P be a hereditary class of bipartite graphs
Problem(P). Given a bipartite graph G, find a
maximum induced subgraph of G belonging to P.
Theorem. If P is a non-trivial hereditary class
of bipartite graphs containing one of the three
minimal factorial classes of bipartite graphs,
then Problem(P) is NP-hard. Otherwise, it is
solvable in polynomial time.
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Thank you