reducing the chromatic number by vertex or edge deletions · an edge is critical if and only if it...
TRANSCRIPT
![Page 1: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/1.jpg)
Reducing the Chromatic Number by Vertex or EdgeDeletions
Authors:Daniel Paulusma (Durham University), Christophe Picouleau (CNAM-PARIS), Bernard Ries
(University of Fribourg)
September 14st 2017
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![Page 2: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/2.jpg)
Definitions
Let G = (V ,E ) be an undirected graph.
A k-coloring of G is a mapping c : V ⇒ {1, 2, . . . k} such thatc(u) 6= c(v) for all uv ∈ E .
The chromatic number of G is the smallest integer k such that Gadmits a k-coloring. It is denoted by χ(G ).
B.C.D. () Chromatic-EV-Deletion September 14st 2017 2 / 23
![Page 3: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/3.jpg)
Definitions
Let G = (V ,E ) be an undirected graph.
A k-coloring of G is a mapping c : V ⇒ {1, 2, . . . k} such thatc(u) 6= c(v) for all uv ∈ E .
The chromatic number of G is the smallest integer k such that Gadmits a k-coloring. It is denoted by χ(G ).
B.C.D. () Chromatic-EV-Deletion September 14st 2017 2 / 23
![Page 4: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/4.jpg)
Definitions
Let G = (V ,E ) be an undirected graph.
A k-coloring of G is a mapping c : V ⇒ {1, 2, . . . k} such thatc(u) 6= c(v) for all uv ∈ E .
The chromatic number of G is the smallest integer k such that Gadmits a k-coloring. It is denoted by χ(G ).
B.C.D. () Chromatic-EV-Deletion September 14st 2017 2 / 23
![Page 5: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/5.jpg)
Definitions
The contraction of an edge uv in G removes the vertices u and v fromG , and replaces them by a new vertex made adjacent to precisely thosevertices that were adjacent to u or v in G .
u
v
a
bc
d
e
f
a
bc
d
e
f
⇒
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Definitions
A vertex or edge is critical if its removal reduces χ(G ) by 1.
An edge is contraction-critical if its contraction reduces χ(G ) by 1.
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Definitions
A vertex or edge is critical if its removal reduces χ(G ) by 1.
An edge is contraction-critical if its contraction reduces χ(G ) by 1.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 4 / 23
![Page 8: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/8.jpg)
Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
![Page 9: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/9.jpg)
Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
![Page 10: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/10.jpg)
Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1.
Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
![Page 11: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/11.jpg)
Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.
Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
![Page 12: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/12.jpg)
Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors.
Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
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Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
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Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.
Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
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Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.
So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
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Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).
Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
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Result
Proposition
An edge is critical if and only if it is contraction-critical.
Proof.
Let e = uv ∈ E .
⇒ e is critical: χ(G − e) = χ(G )− 1. Then u and v are colored alikein any coloring of G − e with χ(G − e) colors.Hence G ′ obtainedfrom contracting e in G can also be colored with χ(G − e) colors. Asχ(G − e) = χ(G )− 1, e is contraction-critical.
⇐ e is contraction-critical: G ′ obtained from contracting e hasχ(G ′) = χ(G )− 1.Coloring u and v alike, we obtain a coloring ofG − e from any coloring of G ′.So χ(G − e) ≤ χ(G ′).Asχ(G ′) = χ(G )− 1, e is critical.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 5 / 23
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Problems
Critical VertexI: A graph G = (V ,E ).Q: Is there a critical vertex ?
Critical EdgeI: A graph G = (V ,E ).Q: Is there a critical edge ?
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Problems
Critical VertexI: A graph G = (V ,E ).Q: Is there a critical vertex ?
Critical EdgeI: A graph G = (V ,E ).Q: Is there a critical edge ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 6 / 23
![Page 20: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/20.jpg)
Result
Theorem
Critical Vertex and Critical Edge are both co-NP-hard for(C5, 4P1, 2P1 + P2, 2P2)-free graphs.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 7 / 23
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Proof
From a restriction of Monotone 1-in-3-SAT which is NP-complete.(C. Moore and J. M. Robson (2001) )
m clauses,
n variables,
each clause consists of three distinct positive literals,
each variable occurs in exactly three clauses.
Note that m = nIs there a truth assignment such that each clause is satisfied by exactlyone variable ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 8 / 23
![Page 22: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/22.jpg)
Proof
From a restriction of Monotone 1-in-3-SAT which is NP-complete.(C. Moore and J. M. Robson (2001) )
m clauses,
n variables,
each clause consists of three distinct positive literals,
each variable occurs in exactly three clauses.
Note that m = nIs there a truth assignment such that each clause is satisfied by exactlyone variable ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 8 / 23
![Page 23: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/23.jpg)
Proof
From a restriction of Monotone 1-in-3-SAT which is NP-complete.(C. Moore and J. M. Robson (2001) )
m clauses,
n variables,
each clause consists of three distinct positive literals,
each variable occurs in exactly three clauses.
Note that m = nIs there a truth assignment such that each clause is satisfied by exactlyone variable ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 8 / 23
![Page 24: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/24.jpg)
Proof
From a restriction of Monotone 1-in-3-SAT which is NP-complete.(C. Moore and J. M. Robson (2001) )
m clauses,
n variables,
each clause consists of three distinct positive literals,
each variable occurs in exactly three clauses.
Note that m = nIs there a truth assignment such that each clause is satisfied by exactlyone variable ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 8 / 23
![Page 25: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/25.jpg)
Proof
From a restriction of Monotone 1-in-3-SAT which is NP-complete.(C. Moore and J. M. Robson (2001) )
m clauses,
n variables,
each clause consists of three distinct positive literals,
each variable occurs in exactly three clauses.
Note that m = nIs there a truth assignment such that each clause is satisfied by exactlyone variable ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 8 / 23
![Page 26: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/26.jpg)
Proof
From a restriction of Monotone 1-in-3-SAT which is NP-complete.(C. Moore and J. M. Robson (2001) )
m clauses,
n variables,
each clause consists of three distinct positive literals,
each variable occurs in exactly three clauses.
Note that m = n
Is there a truth assignment such that each clause is satisfied by exactlyone variable ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 8 / 23
![Page 27: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/27.jpg)
Proof
From a restriction of Monotone 1-in-3-SAT which is NP-complete.(C. Moore and J. M. Robson (2001) )
m clauses,
n variables,
each clause consists of three distinct positive literals,
each variable occurs in exactly three clauses.
Note that m = nIs there a truth assignment such that each clause is satisfied by exactlyone variable ?
B.C.D. () Chromatic-EV-Deletion September 14st 2017 8 / 23
![Page 28: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/28.jpg)
Proof
Consider the Critical Vertex problem.
We prove that the problem of deciding whether a graph has a vertexwhose deletion reduces σ(G ) the clique-covering number by 1 isco-NP-hard for (C4,C5,K4, 2P1 + P2)-free graphs.
The complements of (C4,C5,K4, 2P1 + P2)-free graphs are(C5, 4P1, 2P1 + P2, 2P2)-free
The two problems are equivalent.
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Proof
Consider the Critical Vertex problem.
We prove that the problem of deciding whether a graph has a vertexwhose deletion reduces σ(G ) the clique-covering number by 1 isco-NP-hard for (C4,C5,K4, 2P1 + P2)-free graphs.
The complements of (C4,C5,K4, 2P1 + P2)-free graphs are(C5, 4P1, 2P1 + P2, 2P2)-free
The two problems are equivalent.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 9 / 23
![Page 30: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/30.jpg)
Proof
Consider the Critical Vertex problem.
We prove that the problem of deciding whether a graph has a vertexwhose deletion reduces σ(G ) the clique-covering number by 1 isco-NP-hard for (C4,C5,K4, 2P1 + P2)-free graphs.
The complements of (C4,C5,K4, 2P1 + P2)-free graphs are(C5, 4P1, 2P1 + P2, 2P2)-free
The two problems are equivalent.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 9 / 23
![Page 31: Reducing the Chromatic Number by Vertex or Edge Deletions · An edge is critical if and only if it is contraction-critical. Proof. Let e = uv 2E.)e is critical: ˜(G e) = ˜(G) 1](https://reader034.vdocument.in/reader034/viewer/2022050612/5fb316215f64f04b7c7479db/html5/thumbnails/31.jpg)
Proof
Consider the Critical Vertex problem.
We prove that the problem of deciding whether a graph has a vertexwhose deletion reduces σ(G ) the clique-covering number by 1 isco-NP-hard for (C4,C5,K4, 2P1 + P2)-free graphs.
The complements of (C4,C5,K4, 2P1 + P2)-free graphs are(C5, 4P1, 2P1 + P2, 2P2)-free
The two problems are equivalent.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 9 / 23
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Proof
Construction of G = (V ,E ) from a boolean formula Φ
c(x)
c(z)
c(y)
a1c a2
c
a3ca4
cc'(x)
c"(x)
c'(z) c''(z)
c'(y)
c''(y)
Each clause C : a 7-cycle
Each variable x : a triangle
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Proof
c(x)
c(z)
c(y)
a1c a2
c
a3ca4
cc'(x)
c"(x)
c'(z) c''(z)
c'(y)
c''(y)
|V | = 7n = 7m
(C4,C5,K4, 2P1 + P2)-free
σ(G ) ≥ 10
3n
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Proof
c(x)
c(z)
c(y)
a1c a2
c
a3ca4
cc'(x)
c"(x)
c'(z) c''(z)
c'(y)
c''(y)
Φ is satisfiable if and only if σ(G ) =10
3n;
If G has a clique cover K = {K1, . . . ,K 103
n}, then each Ki ∈ Kconsists of either two or three vertices (no isolated vertices);
If σ(G ) >10
3n, then G has a minimum clique cover K that contains
a clique of size 1.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 12 / 23
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Proof
c(x)
c(z)
c(y)
a1c a2
c
a3ca4
cc'(x)
c"(x)
c'(z) c''(z)
c'(y)
c''(y)
Φ is satisfiable if and only if σ(G ) =10
3n;
If G has a clique cover K = {K1, . . . ,K 103
n}, then each Ki ∈ Kconsists of either two or three vertices (no isolated vertices);
If σ(G ) >10
3n, then G has a minimum clique cover K that contains
a clique of size 1.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 12 / 23
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Proof
c(x)
c(z)
c(y)
a1c a2
c
a3ca4
cc'(x)
c"(x)
c'(z) c''(z)
c'(y)
c''(y)
Φ is satisfiable if and only if σ(G ) =10
3n;
If G has a clique cover K = {K1, . . . ,K 103
n}, then each Ki ∈ Kconsists of either two or three vertices (no isolated vertices);
If σ(G ) >10
3n, then G has a minimum clique cover K that contains
a clique of size 1.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 12 / 23
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Proof
It follows:
If Φ is a yes-instance there is no vertex u with σ(G − u) ≤ σ(G )− 1;
If Φ is a no-instance there exists a vertex u with σ(G −u) = σ(G )−1.
Critical Vertex is co-NP-hard for (C5, 4P1, 2P1 + P2, 2P2)-free graphs.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 13 / 23
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Proof
It follows:
If Φ is a yes-instance there is no vertex u with σ(G − u) ≤ σ(G )− 1;
If Φ is a no-instance there exists a vertex u with σ(G −u) = σ(G )−1.
Critical Vertex is co-NP-hard for (C5, 4P1, 2P1 + P2, 2P2)-free graphs.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 13 / 23
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Proof
It follows:
If Φ is a yes-instance there is no vertex u with σ(G − u) ≤ σ(G )− 1;
If Φ is a no-instance there exists a vertex u with σ(G −u) = σ(G )−1.
Critical Vertex is co-NP-hard for (C5, 4P1, 2P1 + P2, 2P2)-free graphs.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 13 / 23
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Proof
It follows:
If Φ is a yes-instance there is no vertex u with σ(G − u) ≤ σ(G )− 1;
If Φ is a no-instance there exists a vertex u with σ(G −u) = σ(G )−1.
Critical Vertex is co-NP-hard for (C5, 4P1, 2P1 + P2, 2P2)-free graphs.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 13 / 23
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Proof
For the Critical Edge problem:
Slight modification in the construction:
change each C7 into C11
consider clique of size two (edges) instead of clique of size one (vertex)
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Proof
For the Critical Edge problem:
Slight modification in the construction:
change each C7 into C11
consider clique of size two (edges) instead of clique of size one (vertex)
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Proof
For the Critical Edge problem:
Slight modification in the construction:
change each C7 into C11
consider clique of size two (edges) instead of clique of size one (vertex)
B.C.D. () Chromatic-EV-Deletion September 14st 2017 14 / 23
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Proof
For the Critical Edge problem:
Slight modification in the construction:
change each C7 into C11
consider clique of size two (edges) instead of clique of size one (vertex)
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Result
Theorem
Critical Vertex, Critical Edge and Contraction-Critical Edge restricted toH-free graphs are polynomial-time solvable if H ⊆i P1 + P3 or H ⊆i P4,and NP-hard or co-NP-hard otherwise.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 15 / 23
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Proof
Theorem : If a graph H ⊆i P4 or H ⊆i P1 + P3, then Coloring ispolynomial-time solvable for H-free graphs, otherwise it is NP-complete.D. Kral’, J. Kratochvıl, Z. Tuza, and G.J. Woeginger (2001)
H ⊆i P1 + P3 or H ⊆i P4: Let G be H-free.Test if there exists v ∈ V such that χ(G − v) < χ(G ).
(P1 + P3)-free and P4-free graphs are closed under edge contraction.same approach for Contraction-Critical Edge, and forCritical Edge.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 16 / 23
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Proof
Theorem : If a graph H ⊆i P4 or H ⊆i P1 + P3, then Coloring ispolynomial-time solvable for H-free graphs, otherwise it is NP-complete.D. Kral’, J. Kratochvıl, Z. Tuza, and G.J. Woeginger (2001)
H ⊆i P1 + P3 or H ⊆i P4:
Let G be H-free.Test if there exists v ∈ V such that χ(G − v) < χ(G ).
(P1 + P3)-free and P4-free graphs are closed under edge contraction.same approach for Contraction-Critical Edge, and forCritical Edge.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 16 / 23
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Proof
Theorem : If a graph H ⊆i P4 or H ⊆i P1 + P3, then Coloring ispolynomial-time solvable for H-free graphs, otherwise it is NP-complete.D. Kral’, J. Kratochvıl, Z. Tuza, and G.J. Woeginger (2001)
H ⊆i P1 + P3 or H ⊆i P4: Let G be H-free.
Test if there exists v ∈ V such that χ(G − v) < χ(G ).
(P1 + P3)-free and P4-free graphs are closed under edge contraction.same approach for Contraction-Critical Edge, and forCritical Edge.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 16 / 23
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Proof
Theorem : If a graph H ⊆i P4 or H ⊆i P1 + P3, then Coloring ispolynomial-time solvable for H-free graphs, otherwise it is NP-complete.D. Kral’, J. Kratochvıl, Z. Tuza, and G.J. Woeginger (2001)
H ⊆i P1 + P3 or H ⊆i P4: Let G be H-free.Test if there exists v ∈ V such that χ(G − v) < χ(G ).
(P1 + P3)-free and P4-free graphs are closed under edge contraction.same approach for Contraction-Critical Edge, and forCritical Edge.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 16 / 23
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Proof
Theorem : If a graph H ⊆i P4 or H ⊆i P1 + P3, then Coloring ispolynomial-time solvable for H-free graphs, otherwise it is NP-complete.D. Kral’, J. Kratochvıl, Z. Tuza, and G.J. Woeginger (2001)
H ⊆i P1 + P3 or H ⊆i P4: Let G be H-free.Test if there exists v ∈ V such that χ(G − v) < χ(G ).
(P1 + P3)-free and P4-free graphs are closed under edge contraction.
same approach for Contraction-Critical Edge, and forCritical Edge.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 16 / 23
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Proof
Theorem : If a graph H ⊆i P4 or H ⊆i P1 + P3, then Coloring ispolynomial-time solvable for H-free graphs, otherwise it is NP-complete.D. Kral’, J. Kratochvıl, Z. Tuza, and G.J. Woeginger (2001)
H ⊆i P1 + P3 or H ⊆i P4: Let G be H-free.Test if there exists v ∈ V such that χ(G − v) < χ(G ).
(P1 + P3)-free and P4-free graphs are closed under edge contraction.same approach for Contraction-Critical Edge,
and forCritical Edge.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 16 / 23
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Proof
Theorem : If a graph H ⊆i P4 or H ⊆i P1 + P3, then Coloring ispolynomial-time solvable for H-free graphs, otherwise it is NP-complete.D. Kral’, J. Kratochvıl, Z. Tuza, and G.J. Woeginger (2001)
H ⊆i P1 + P3 or H ⊆i P4: Let G be H-free.Test if there exists v ∈ V such that χ(G − v) < χ(G ).
(P1 + P3)-free and P4-free graphs are closed under edge contraction.same approach for Contraction-Critical Edge, and forCritical Edge.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 16 / 23
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Proof
H 6⊆i P1 + P3 and H 6⊆i P4:
Theorem : Critical Vertex and Critical Edge are NP-hardfor H-free graphs if H ⊇i K1,3 or H ⊇i Cr for some r ≥ 3.D. Paulusma, Ch. P. and B. Ries (2016)
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Proof
H 6⊆i P1 + P3 and H 6⊆i P4:Theorem : Critical Vertex and Critical Edge are NP-hardfor H-free graphs if H ⊇i K1,3 or H ⊇i Cr for some r ≥ 3.D. Paulusma, Ch. P. and B. Ries (2016)
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Proof
H 6⊆i P1 + P3 and H 6⊆i P4:Theorem : Critical Vertex and Critical Edge are NP-hardfor H-free graphs if H ⊇i K1,3 or H ⊇i Cr for some r ≥ 3.D. Paulusma, Ch. P. and B. Ries (2016)
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Proof
H is a disjoint union of r paths for some r ≥ 1:
I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for(C5, 4P1, 2P1 + P2, 2P2)-free graphs.
I r = 3:F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4:
we proved Critical Vertex and Critical Edge are co-NP-hard for(C5, 4P1, 2P1 + P2, 2P2)-free graphs.
I r = 3:F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.
I r = 3:F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;
F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:
F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
B.C.D. () Chromatic-EV-Deletion September 14st 2017 18 / 23
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result above
F H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4then 2P1 + P2 ⊆i H: use the result above
F r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use theresult above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk :
if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4then 2P1 + P2 ⊆i H: use the result above
F r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use theresult above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded.
If k ≥ 4then 2P1 + P2 ⊆i H: use the result above
F r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use theresult above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result above
F r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use theresult above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk :
k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use theresult above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded,
then k ≥ 5 so 2P2 ⊆i H and use theresult above.
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Proof
H is a disjoint union of r paths for some r ≥ 1:I r ≥ 4: we proved Critical Vertex and Critical Edge are co-NP-hard for
(C5, 4P1, 2P1 + P2, 2P2)-free graphs.I r = 3:
F H = 3P1 ⊆i P1 + P3: case excluded;F H ⊇i 2P1 + P2: use the result above
I r = 2:F both paths contain an edge: H ⊇i 2P2, use the result aboveF H ⊆i P1 + Pk : if k ≤ 3 then H ⊆i P1 + P3: case excluded. If k ≥ 4
then 2P1 + P2 ⊆i H: use the result aboveF r = 1, H = Pk : k ≤ 4 is excluded, then k ≥ 5 so 2P2 ⊆i H and use the
result above.
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Proof
Since Critical Edge and Contraction-Critical Edge are equivalent we havethe dichotomy:
Critical Vertex, Critical Edge and Contraction-Critical Edge restricted toH-free graphs are polynomial-time solvable if H ⊆i P1 + P3 or H ⊆i P4,and NP-hard or co-NP-hard otherwise.
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Proof
Since Critical Edge and Contraction-Critical Edge are equivalent we havethe dichotomy:
Critical Vertex, Critical Edge and Contraction-Critical Edge restricted toH-free graphs
are polynomial-time solvable if H ⊆i P1 + P3 or H ⊆i P4,and NP-hard or co-NP-hard otherwise.
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Proof
Since Critical Edge and Contraction-Critical Edge are equivalent we havethe dichotomy:
Critical Vertex, Critical Edge and Contraction-Critical Edge restricted toH-free graphs are polynomial-time solvable if H ⊆i P1 + P3 or H ⊆i P4,
and NP-hard or co-NP-hard otherwise.
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Proof
Since Critical Edge and Contraction-Critical Edge are equivalent we havethe dichotomy:
Critical Vertex, Critical Edge and Contraction-Critical Edge restricted toH-free graphs are polynomial-time solvable if H ⊆i P1 + P3 or H ⊆i P4,and NP-hard or co-NP-hard otherwise.
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Blocker
Contraction Blocker(χ)I: G = (V ,E ) and two integers d , k ≥ 0Q: can G be k-contracted into G ′ such that χ(G ′) ≤ χ(G )− d ?
d = k = 1 corresponds to Contraction-Critical Edge
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Blocker
Contraction Blocker(χ)I: G = (V ,E ) and two integers d , k ≥ 0Q: can G be k-contracted into G ′ such that χ(G ′) ≤ χ(G )− d ?
d = k = 1 corresponds to Contraction-Critical Edge
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Blocker
Contraction Blocker(χ)I: G = (V ,E ) and two integers d , k ≥ 0Q: can G be k-contracted into G ′ such that χ(G ′) ≤ χ(G )− d ?
d = k = 1 corresponds to Contraction-Critical Edge
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Blocker
Vertex Deletion Blocker(χ)I: G = (V ,E ) and two integers d , k ≥ 0Q: can G be k-vertex-deleted into G ′ such that χ(G ′) ≤ χ(G )− d ?
d = k = 1 corresponds to Critical Vertex
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Blocker
Vertex Deletion Blocker(χ)I: G = (V ,E ) and two integers d , k ≥ 0Q: can G be k-vertex-deleted into G ′ such that χ(G ′) ≤ χ(G )− d ?
d = k = 1 corresponds to Critical Vertex
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Blocker
Theorem
If H ⊆i P4 Contraction Blocker(χ) is polynomial-time solvablefor H-free graphs, it is NP-hard otherwise.
If H ⊆i P1 + P3 or P4 Vertex Deletion Blocker(χ) for H-freegraphs is polynomial-time solvable, it is NP-hard or co-NP-hardotherwise.
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Blocker
Theorem
If H ⊆i P4 Contraction Blocker(χ) is polynomial-time solvablefor H-free graphs, it is NP-hard otherwise.
If H ⊆i P1 + P3 or P4 Vertex Deletion Blocker(χ) for H-freegraphs is polynomial-time solvable, it is NP-hard or co-NP-hardotherwise.
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Merci beaucoup pour votre attention
B.C.D. () Chromatic-EV-Deletion September 14st 2017 23 / 23