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