between 2- and 3-colorability rutgers university
TRANSCRIPT
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Between 2- and 3-colorability
Rutgers University
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The problem
GX Bipartite
Graph
O Independent Set
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The problem
GX Bipartite
Graph• tree
O Independent Set
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The problem
GX Bipartite
Graph• tree• forest
O Independent Set
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The problem
GX Bipartite
Graph• tree• forest• of bounded degree
O Independent Set
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The problem
GX Bipartite
Graph• tree• forest• of bounded degree• complete bipartite
O Independent Set
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Examples
• Trees NP-completeA. 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-completeA. 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-completeA. 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-completeJ. 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-completeA. 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-completeJ. Kratochvíl, I. Schiermeyer, On the computational complexity of (O, P)-partition problems. Discuss. Math. Graph Theory 17 (1997) 253--258.
• Complete bipartite PolynomialA. 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 XP implies X-vP for any vertex vV(X)
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Hereditary classes of graphs
Definition.
A class of graphs P is hereditary if XP implies X-vP for any vertex vV(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
permutation graphs line graphs
graphs of bounded vertex degree graphs of bounded tree-width
planar graphs
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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 graphsProblem. 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 graphsProblem. 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 graphsProblem*(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 graphsProblem*(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