On the Complexity of Defective Coloring Rmy Belmonte 1 Michael Lampis - - PowerPoint PPT Presentation

1/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

On the Complexity of Defective Coloring

Rémy Belmonte1 Michael Lampis2 Valia Mitsou3

1University of Electro-Communications, Tokyo 2Lamsade, CNRS and Université Paris-Dauphine, Paris 3Liris, CNRS and Université Lyon 1, Lyon

June 20, 2017

2/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Definitions and motivation

3/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

(Defective) Coloring

Definition (Proper coloring) A proper (vertex) c-coloring of a graph G is an assignment of colors {1, . . . , c} to the vertices of G such that no two adjacent vertices are assigned the same color.

Figure: Petersen graph

3/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

(Defective) Coloring

Definition (Proper coloring) A proper (vertex) c-coloring of a graph G is an assignment of colors {1, . . . , c} to the vertices of G such that no two adjacent vertices are assigned the same color.

Figure: A proper 3-coloring

3/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

(Defective) Coloring

Definition (Defective coloring) A defective (c, ∆∗)-coloring of a graph G is an assignment of colors {1, . . . , c} to the vertices of G such that every vertex of G has at most ∆∗ neighbors sharing its color.

Figure: A defective (2,1)-coloring

4/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Applications: Radio networks

Antennas which are within each-other’s range must be assigned different frequencies.

4/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Applications: Radio networks

Antennas which are within each-other’s range must be assigned different frequencies.

4/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Applications: Radio networks

Antennas which are within each-other’s range must be assigned different frequencies. If we can afford signal interference from at most ∆∗ = 1 neighboring antenna, we can use less frequencies.

5/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - simple facts

For any graph G with n vertices: A proper c-coloring of G is a defective (c, 0)-coloring of G.

5/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - simple facts

For any graph G with n vertices: A proper c-coloring of G is a defective (c, 0)-coloring of G. Defective coloring is already NP-hard for c = 3 and ∆∗ = 0.

5/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - simple facts

For any graph G with n vertices: A proper c-coloring of G is a defective (c, 0)-coloring of G. Defective coloring is already NP-hard for c = 3 and ∆∗ = 0. There always exists an (1, ∆)-coloring of G, where ∆ is its maximum degree.

5/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - simple facts

For any graph G with n vertices: A proper c-coloring of G is a defective (c, 0)-coloring of G. Defective coloring is already NP-hard for c = 3 and ∆∗ = 0. There always exists an (1, ∆)-coloring of G, where ∆ is its maximum degree. If c · k ≥ n then G has a (c, k − 1)-coloring.

Figure: Any graph on n vertices is (c, ⌈n/c⌉ − 1)-colorable

6/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - previous work

Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986].

6/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - previous work

Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]:

6/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - previous work

Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]:

For any ∆∗ ≥ 1, (2, ∆∗)-coloring is NP-complete on planar graphs.

6/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - previous work

Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]:

For any ∆∗ ≥ 1, (2, ∆∗)-coloring is NP-complete on planar graphs. (3, 1)-coloring is NP-complete on planar graphs.

6/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - previous work

Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]:

For any ∆∗ ≥ 1, (2, ∆∗)-coloring is NP-complete on planar graphs. (3, 1)-coloring is NP-complete on planar graphs.

[Havet, Kang, and Sereni 2009]: For any c ≥ 2, ∆∗ ≥ 1, (c, ∆∗)-coloring is NP-complete on unit-disk graphs.

6/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective Coloring - previous work

Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]:

For any ∆∗ ≥ 1, (2, ∆∗)-coloring is NP-complete on planar graphs. (3, 1)-coloring is NP-complete on planar graphs.

[Havet, Kang, and Sereni 2009]: For any c ≥ 2, ∆∗ ≥ 1, (c, ∆∗)-coloring is NP-complete on unit-disk graphs. [Angelini et al. 2017]: (3, 1)-coloring is NP-complete on graphs of degree at most 6 and on planar graphs of degree at most 7.

7/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Coloring Perfect Graphs

Definition A graph G is called perfect when for every induced subgraph, its chromatic number equals its clique number.

7/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Coloring Perfect Graphs

Definition A graph G is called perfect when for every induced subgraph, its chromatic number equals its clique number. Theorem (Proper) coloring is polynomially solvable on perfect graphs.

7/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Coloring Perfect Graphs

Definition A graph G is called perfect when for every induced subgraph, its chromatic number equals its clique number. Theorem (Proper) coloring is polynomially solvable on perfect graphs. Motivation Study defective coloring on classes of perfect graphs.

7/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Coloring Perfect Graphs

Chordal graphs Cographs NP-hard on Split if c ≥ 2 NP-hard NP-hard on Split if ∆∗ ≥ 1 In P if c or ∆∗ is fixed In P if c, ∆∗ fixed Solvable in n

O

• n4

/5

In P on Trivially perfect for any c, ∆∗

8/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs

9/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - hardness

Definition A graph G is chordal if all cycles of length ≥ 4 have a chord.

9/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - hardness

Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. Definition A graph G is split if its vertices can be partitioned into a clique and an independent set.

9/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - hardness

Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. Definition A graph G is split if its vertices can be partitioned into a clique and an independent set. Fact Split graphs are chordal.

9/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - hardness

Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. Definition A graph G is split if its vertices can be partitioned into a clique and an independent set. Fact Split graphs are chordal. Theorem Defective coloring is NP-hard on split graphs for any ∆∗ ≥ 1.

9/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - hardness

Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. Definition A graph G is split if its vertices can be partitioned into a clique and an independent set. Fact Split graphs are chordal. Theorem Defective coloring is NP-hard on split graphs for any ∆∗ ≥ 1. Theorem Defective coloring is NP-hard on split graphs for any c ≥ 2.

10/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2

We first create an equality gadget that forces two vertices u, v to receive the same color:

10/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2

We first create an equality gadget that forces two vertices u, v to receive the same color:

construct an independent set of size 2∆∗ + 1; construct vertices u, v connecting to the independent set.

10/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2

We first create an equality gadget that forces two vertices u, v to receive the same color:

construct an independent set of size 2∆∗ + 1; construct vertices u, v connecting to the independent set. One of the two colors (say red) should be a majority in the IS, with at least ∆∗ + 1 red vertices.

10/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2

We first create an equality gadget that forces two vertices u, v to receive the same color:

construct an independent set of size 2∆∗ + 1; construct vertices u, v connecting to the independent set. One of the two colors (say red) should be a majority in the IS, with at least ∆∗ + 1 red vertices. None of u, v can be colored red, thus they should both be blue.

10/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2

We first create an equality gadget that forces two vertices u, v to receive the same color:

construct an independent set of size 2∆∗ + 1; construct vertices u, v connecting to the independent set. One of the two colors (say red) should be a majority in the IS, with at least ∆∗ + 1 red vertices. None of u, v can be colored red, thus they should both be blue. Recoloring the entire IS red restores the capacity of u, v to ∆∗.

10/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2

We first create an equality gadget that forces two vertices u, v to receive the same color:

construct an independent set of size 2∆∗ + 1; construct vertices u, v connecting to the independent set. One of the two colors (say red) should be a majority in the IS, with at least ∆∗ + 1 red vertices. None of u, v can be colored red, thus they should both be blue. Recoloring the entire IS red restores the capacity of u, v to ∆∗.

Connecting u, v decreases their capacity by 1. Placing an entire clique of size q decreases the capacity to ∆∗ − (q − 1).

10/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2

We first create an equality gadget that forces two vertices u, v to receive the same color:

construct an independent set of size 2∆∗ + 1; construct vertices u, v connecting to the independent set. One of the two colors (say red) should be a majority in the IS, with at least ∆∗ + 1 red vertices. None of u, v can be colored red, thus they should both be blue. Recoloring the entire IS red restores the capacity of u, v to ∆∗.

Connecting u, v decreases their capacity by 1. Placing an entire clique of size q decreases the capacity to ∆∗ − (q − 1). The equality gadget is a split graph.

11/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.”

11/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” Construction: Construct the incidence graph (X, C).

11/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” Construction: Construct the incidence graph (X, C). Turn C into a clique.

11/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” Construction: Construct the incidence graph (X, C). Turn C into a clique. Add a true twin clique C′ of C.

11/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” Construction: Construct the incidence graph (X, C). Turn C into a clique. Add a true twin clique C′ of C. Construct two independent sets Z, Z ′, each of size 2m + 3, connecting to C, C′ respectively.

11/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” Construction: Construct the incidence graph (X, C). Turn C into a clique. Add a true twin clique C′ of C. Construct two independent sets Z, Z ′, each of size 2m + 3, connecting to C, C′ respectively. → The formula is satisfiable iff G admits a (2, m + 1)-coloring.

12/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” → The formula is satisfiable iff G admits a (2, m + 1)-coloring. Proof.

12/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” → The formula is satisfiable iff G admits a (2, m + 1)-coloring. Proof.

(C, Z), (C′, Z ′) form eq. gadgets.

12/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” → The formula is satisfiable iff G admits a (2, m + 1)-coloring. Proof.

(C, Z), (C′, Z ′) form eq. gadgets. ⇒ All vertices in C have the same color (say blue). Also, all vertices in C′ have the same color.

12/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” → The formula is satisfiable iff G admits a (2, m + 1)-coloring. Proof.

(C, Z), (C′, Z ′) form eq. gadgets. ⇒ All vertices in C have the same color (say blue). Also, all vertices in C′ have the same color. The capacity of the vertices in C is now m + 1 − (m − 1) = 2, thus the colors of C, C′ should be different.

12/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” → The formula is satisfiable iff G admits a (2, m + 1)-coloring. Proof.

(C, Z), (C′, Z ′) form eq. gadgets. ⇒ All vertices in C have the same color (say blue). Also, all vertices in C′ have the same color. The capacity of the vertices in C is now m + 1 − (m − 1) = 2, thus the colors of C, C′ should be different. A correct truth assignment implies a good coloring. .

12/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Split graphs: hardness for c = 2 (cont.)

Start from 3NAESAT: “Given positive CNF (n variables, m clauses), assign T, F such that every clause contains at least one of each.” → The formula is satisfiable iff G admits a (2, m + 1)-coloring. Proof.

(C, Z), (C′, Z ′) form eq. gadgets. ⇒ All vertices in C have the same color (say blue). Also, all vertices in C′ have the same color. The capacity of the vertices in C is now m + 1 − (m − 1) = 2, thus the colors of C, C′ should be different. A correct truth assignment implies a good coloring. If there is no correct assignment, we can find a vertex in C or C′ with deficiency m + 2.

13/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - algorithms

Theorem Defective coloring is FPT on chordal graphs if both ∆∗ and c are parameters.

13/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - algorithms

Theorem Defective coloring is FPT on chordal graphs if both ∆∗ and c are parameters. Proof. (Fact: For a chordal graph G, ω(G) = tw(G) + 1.)

13/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - algorithms

Theorem Defective coloring is FPT on chordal graphs if both ∆∗ and c are parameters. Proof. (Fact: For a chordal graph G, ω(G) = tw(G) + 1.) If ω(G) > c(∆∗ + 1) then the answer is “no”.

13/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Chordal Graphs - algorithms

Theorem Defective coloring is FPT on chordal graphs if both ∆∗ and c are parameters. Proof. (Fact: For a chordal graph G, ω(G) = tw(G) + 1.) If ω(G) > c(∆∗ + 1) then the answer is “no”. Else ω = tw + 1 ≤ c(∆∗ + 1).

(DP for treewidth runs in (∆∗ · c)O(tw) · nO(1).)

14/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs

15/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: definition and properties

Definition G is a cograph if it can be constructed from single vertices by a sequence of join and disjoint union operations.

15/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: definition and properties

Definition G is a cograph if it can be constructed from single vertices by a sequence of join and disjoint union operations.

15/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: definition and properties

Definition G is a cograph if it can be constructed from single vertices by a sequence of join and disjoint union operations.

15/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: definition and properties

Definition G is a cograph if it can be constructed from single vertices by a sequence of join and disjoint union operations. Fact Cographs are exactly graphs of clique-width 2.

16/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: proper coloring

Disjoint union: |c(V(G1 ⊕ G2))| = max{(|c(V(G1))|, |c(V(G2))|}.

16/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: proper coloring

Disjoint union: |c(V(G1 ⊕ G2))| = max{(|c(V(G1))|, |c(V(G2))|}.

16/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: proper coloring

Disjoint union: |c(V(G1 ⊕ G2))| = max{(|c(V(G1))|, |c(V(G2))|}. Join: |c(V(G1 + G2))| = |c(V(G1))| + |c(V(G2))|.

16/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: proper coloring

Disjoint union: |c(V(G1 ⊕ G2))| = max{(|c(V(G1))|, |c(V(G2))|}. Join: |c(V(G1 + G2))| = |c(V(G1))| + |c(V(G2))|.

17/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: results

Theorem Defective Coloring is NP-complete on cographs (actually even on complete multipartite graphs).

17/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: results

Theorem Defective Coloring is NP-complete on cographs (actually even on complete multipartite graphs). Theorem Defective Coloring can be solved in polynomial time on cographs when either c or ∆∗ is fixed.

17/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: results

Theorem Defective Coloring is NP-complete on cographs (actually even on complete multipartite graphs). Theorem Defective Coloring can be solved in polynomial time on cographs when either c or ∆∗ is fixed. Theorem There exists a sub-exponential time algorithm for Defective Coloring

• n cographs.

18/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: hardness

4-Partition Given four sets A1, A2, A3, A4, Ai ⊂ I N, |Ai| = n and a target B ∈ I N, create n quadruples qi ∈ A1 × A2 × A3 × A4 such that each ai ∈ Ai appears exaclty once and the sum of the elements in any qi is B. We reduce from a special version of 4-Partition, where elements of Ai have sorted values such that for each i, min{ai ∈ Ai} > B/2i.

19/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: hardness (cont.)

Construction: For every element ai, create an independent set of ai vertices. For every element in A4 create an additional IS. Interconnect all IS.

19/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: hardness (cont.)

Construction: For every element ai, create an independent set of ai vertices. For every element in A4 create an additional IS. Interconnect all IS. Main idea: If there is a valid partition then there is a (n, B)-coloring: all elements corresponding to quadruple qi receive color i; the deficiency is the degree of the elements in the smallest set.

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming:

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming: DP algorithm I solves the problem in O∗ cO((∆∗)4) .

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming: DP algorithm I solves the problem in O∗ cO((∆∗)4) . DP algorithm II solves the problem in O∗ (∆∗)O(c) .

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming: DP algorithm I solves the problem in O∗ cO((∆∗)4) . DP algorithm II solves the problem in O∗ (∆∗)O(c) . Theorem There is a n

O

• n

4 /5

• time algorithm for Defective Coloring on cographs.

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming: DP algorithm I solves the problem in O∗ cO((∆∗)4) . DP algorithm II solves the problem in O∗ (∆∗)O(c) . Theorem There is a n

O

• n

4 /5

• time algorithm for Defective Coloring on cographs.

Proof.

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming: DP algorithm I solves the problem in O∗ cO((∆∗)4) . DP algorithm II solves the problem in O∗ (∆∗)O(c) . Theorem There is a n

O

• n

4 /5

• time algorithm for Defective Coloring on cographs.

Proof. If c · ∆∗ ≥ n then reply “yes”.

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming: DP algorithm I solves the problem in O∗ cO((∆∗)4) . DP algorithm II solves the problem in O∗ (∆∗)O(c) . Theorem There is a n

O

• n

4 /5

• time algorithm for Defective Coloring on cographs.

Proof. If c · ∆∗ ≥ n then reply “yes”. If ∆∗ < n

1/5 then DPI runs in time n

O

• n

4 /5

.

20/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Cographs: algorithms

We can perform clique-width-based Dynamic Programming: DP algorithm I solves the problem in O∗ cO((∆∗)4) . DP algorithm II solves the problem in O∗ (∆∗)O(c) . Theorem There is a n

O

• n

4 /5

• time algorithm for Defective Coloring on cographs.

Proof. If c · ∆∗ ≥ n then reply “yes”. If ∆∗ < n

1/5 then DPI runs in time n

O

• n

4 /5

. Else c < n/∆∗ < n

4/5. Then, DPII runs in n

O

• n

4 /5

.

21/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Trivially perfect graphs

22/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Trivially perfect graphs: definition and properties

Definition A graph G is trivially perfect if it can be constructed from single vertices by a sequence of join and disjoint union operations where joins are restricted to have one of the two operands be a single vertex.

22/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Trivially perfect graphs: definition and properties

Definition A graph G is trivially perfect if it can be constructed from single vertices by a sequence of join and disjoint union operations where joins are restricted to have one of the two operands be a single vertex.

22/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Trivially perfect graphs: definition and properties

Definition A graph G is trivially perfect if it can be constructed from single vertices by a sequence of join and disjoint union operations where joins are restricted to have one of the two operands be a single vertex.

22/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Trivially perfect graphs: definition and properties

Definition A graph G is trivially perfect if it can be constructed from single vertices by a sequence of join and disjoint union operations where joins are restricted to have one of the two operands be a single vertex.

22/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Trivially perfect graphs: definition and properties

Definition A graph G is trivially perfect if it can be constructed from single vertices by a sequence of join and disjoint union operations where joins are restricted to have one of the two operands be a single vertex.

23/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective coloring on TP graphs

Theorem DEFECTIVE COLORING can be solved in polynomial time on trivially perfect graphs. (Greedy lexicographic coloring is optimal.)

23/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective coloring on TP graphs

Theorem DEFECTIVE COLORING can be solved in polynomial time on trivially perfect graphs. (Greedy lexicographic coloring is optimal.) Proof (outline) Given an optimal (c, ∆∗)-coloring f we will transform it to a GL (c, ∆∗)-coloring g.

23/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective coloring on TP graphs

Theorem DEFECTIVE COLORING can be solved in polynomial time on trivially perfect graphs. (Greedy lexicographic coloring is optimal.) Proof (outline) Given an optimal (c, ∆∗)-coloring f we will transform it to a GL (c, ∆∗)-coloring g. Perform a heapsort-like sorting to f(V) in order to produce another optimal coloring h.

23/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Defective coloring on TP graphs

Theorem DEFECTIVE COLORING can be solved in polynomial time on trivially perfect graphs. (Greedy lexicographic coloring is optimal.) Proof (outline) Given an optimal (c, ∆∗)-coloring f we will transform it to a GL (c, ∆∗)-coloring g. Perform a heapsort-like sorting to f(V) in order to produce another optimal coloring h. We can prove by induction that g is as good as any sorted

• coloring. Since h is sorted and optimal, g is optimal as well.

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

→ If color f(v) appears somewhere along the path from the root to u,

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

→ If color f(v) appears somewhere along the path from the root to u, exchange colors on vertices u, v.

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

→ If color f(v) appears somewhere along the path from the root to u, exchange colors on vertices u, v. → If color f(v) does not appear on the path from the root to u,

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

→ If color f(v) appears somewhere along the path from the root to u, exchange colors on vertices u, v. → If color f(v) does not appear on the path from the root to u, exchange colors f(v), f(u) anywhere on the subgraph below u.

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

→ If color f(v) appears somewhere along the path from the root to u, exchange colors on vertices u, v. → If color f(v) does not appear on the path from the root to u, exchange colors f(v), f(u) anywhere on the subgraph below u.

The sorting is performed in a heapsort-like manner:

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

→ If color f(v) appears somewhere along the path from the root to u, exchange colors on vertices u, v. → If color f(v) does not appear on the path from the root to u, exchange colors f(v), f(u) anywhere on the subgraph below u.

The sorting is performed in a heapsort-like manner:

find the maximum color on each subtree below the root and if it is larger than the root’s color exchange them;

24/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

GL algorithm - proof

Consider a path in the underlying tree that contains an unsorted pair (f(u), f(v)). Consider the highest node of color f(u).

→ If color f(v) appears somewhere along the path from the root to u, exchange colors on vertices u, v. → If color f(v) does not appear on the path from the root to u, exchange colors f(v), f(u) anywhere on the subgraph below u.

The sorting is performed in a heapsort-like manner:

find the maximum color on each subtree below the root and if it is larger than the root’s color exchange them; continue recursively.

25/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Summary - Open problems

26/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Summary

We studied Defective Coloring on classes of graphs where Proper Coloring is easy:

26/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Summary

We studied Defective Coloring on classes of graphs where Proper Coloring is easy:

Chordal graphs Cographs NP-hard on Split if c ≥ 2 NP-hard NP-hard on Split if ∆∗ ≥ 1 In P if c or ∆∗ is fixed In P if c, ∆∗ fixed Solvable in n

O

• n4

/5

In P on Trivially perfect for any c, ∆∗

Table: Results on perfect graphs

26/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Summary

We studied Defective Coloring on classes of graphs where Proper Coloring is easy:

Chordal graphs Cographs NP-hard on Split if c ≥ 2 NP-hard NP-hard on Split if ∆∗ ≥ 1 In P if c or ∆∗ is fixed In P if c, ∆∗ fixed Solvable in n

O

• n4

/5

In P on Trivially perfect for any c, ∆∗

Table: Results on perfect graphs

Treewidth FVS (tw · ∆∗)O(tw) FPT for c ≥ 3 W-hard W-hard for c = 2

Table: Results on bounded treewidth

27/27 Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems

Open problems

Question 1: What is the complexity of Defective Coloring on interval graphs? Question 2: Approximation?