Between 2- and 3- colorability Rutgers University
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Between 2- and 3-colorability
Rutgers University
The problem
GX Bipartite
Graph
O Independent Set
The problem
GX Bipartite
Graph• tree
O Independent Set
The problem
GX Bipartite
Graph• tree• forest
O Independent Set
The problem
GX Bipartite
Graph• tree• forest• of bounded degree
O Independent Set
The problem
GX Bipartite
Graph• tree• forest• of bounded degree• complete bipartite
O Independent Set
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.
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
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.
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.
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 XP implies X-vP for any vertex vV(X)
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.
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
permutation graphs line graphs
graphs of bounded vertex degree graphs of bounded tree-width
planar graphs
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 graphsProblem. 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 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.
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 graphsProblem*(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 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.
Thank you
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