Between 2- and 3-colorability
Rutgers University
The problem
O
Independent Set
X
Bipartite Graph
G
The problem
O
Independent Set
X
Bipartite Graph
G
• tree
The problem
O
Independent Set
X
Bipartite Graph
G
• tree
• forest
The problem
O
Independent Set
X
Bipartite Graph
G
• tree
• forest
• of bounded degree
The problem
O
Independent Set
X
Bipartite Graph
G
•
•
•
•
tree
forest
of bounded degree
complete bipartite
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.
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
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.
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.
Question
Is there any boundary separating difficult
instances of the (O,P)-partition problem
from polynomially solvable ones?
Question
Is there any boundary separating difficult
instances of the (O,P)-partition problem
from polynomially solvable ones?
Yes ?
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)
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.
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.
Lower Layers
• constant
• polynomial
• exponential
• factorial
Lower Layers
• constant
• polynomial
• exponential
• factorial
 planar graphs
 permutation graphs
 line graphs
 graphs of bounded vertex degree
 graphs of bounded tree-width
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
Three minimal factorial classes
of bipartite graphs
 P1 The class of graphs of vertex degree at most 1
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
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)
(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
(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.
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.
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
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.
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
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.
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.
One more result
Yannakakis, M. Node-deletion problems on bipartite graphs.
SIAM J. Comput. 10 (1981), no. 2, 310--327.
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.
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.
Thank you