Split clique graph complexity

Apresentação em tema: "Split clique graph complexity"— Transcrição da apresentação:

1 Split clique graph complexity
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Split clique graph complexity L. Alcón and M. Gutierrez La Plata, Argentina L. Faria and C. M. H. de Figueiredo, Rio de Janeiro, Brazil Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

2 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph A graph G is the clique graph of a graph H if G is the graph of intersection in vertices of the maximal cliques of H. 2 1 2 1 4 3 H G 4 3 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

3 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Our Problem 1 2 3 4 G 1 2 3 4 H Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

4 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Previous result WG’2006 – CLIQUE GRAPH is NPC for graphs with maximum degree 14 and maximum clique size 12 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

5 G H 1 2 3 4 1 2 3 4 Classes of graphs where CLIQUE GRAPH is polynomial
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Classes of graphs where CLIQUE GRAPH is polynomial 1 2 3 4 G 1 2 3 4 H Survey of Jayme L. Szwarcfiter Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

6 A quest for a non-trivial
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. A quest for a non-trivial Polynomial decidable class Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

7 Chordal? Split? Clique structure, simplicial elimination sequence, … ?
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Chordal? Clique structure, simplicial elimination sequence, … ? Split? Same as chordal, one clique and one independent set … ? Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

8 Our class: Split G=(V,E) is a split graph if V=(K,S), where K is a complete set and S is an independent set

9 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Main used statements Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

10 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. ksplitp Given a pair of integers k > p, G is ksplitp if G is a split (K,S) graph and for every vertex s of S, p < d(s) < k. Example of 3split2 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

11 In this talk 1. |S| < 3 2. |K| < 4 3. s has a private neighbour
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. In this talk Establish 3 subclasses of split (K,S) graphs in P: 1. |S| < 3 2. |K| < 4 3. s has a private neighbour CLIQUE GRAPH is NP-complete for 3split2 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

12 s1 1. |S| < 3 N(s1) s3 s2 N(s2) N(s3) G is clique graph iff
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. s1 1. |S| < 3 N(s2) N(s1) N(s3) s3 s2 G is clique graph iff G is not Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

13 2. |K| < 4 G is clique graph iff G has no bases set with
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. 2. |K| < 4 G is clique graph iff G has no bases set with Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

14 3. s has a private neighbour
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. 3. s has a private neighbour G is a clique graph Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

15 Our NPC problem NPC CLIQUE GRAPH 3split2 3SAT3
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Our NPC problem CLIQUE GRAPH 3split2 INSTANCE: A 3 3split2 graph G=(V,E) with partition (K,S) QUESTION: Is there a graph H such that G=K(H)? NPC 3SAT3 INSTANCE: A set of variables U, a collection of clauses C, s.t. if c of C, then |c|=2 or |c|=3, each positive literal u occurs once, each negative literal occurs once or twice. QUESTION: Is there a truth assignment for U satisfying each clause of C? Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

16 Some preliminaries Black vertices in K White vertices in S
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Some preliminaries Black vertices in K White vertices in S Theorem – K is assumed in every 3split2 RS-family Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

17 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. 3split2 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

18 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. 3split2 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

19 The evil triangle s1 s2 k1 k3 k2 s3
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. The evil triangle s1 s2 k1 k3 k2 s3 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

20 The main gadget Variable
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. The main gadget Variable Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

21 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

22 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

23 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

24 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

25 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

26 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

27 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Variable Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

28 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

29 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

30 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Clause Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

31 Clique graph is NP-complete - Álcon, L. , Faria, L. , de Figueiredo, C
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. I=(U,C)=({u1, u2, u3, u4, u5, u6, u7}, {(u1,~u2), (u2,~u3), (~u1,u4), (~u2 ,~u4 ,~u5), (~u4 ,~u7), (u5 , ~u6 ,u7), (u3 , u6)} Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

32 ~u1=~u2=~u3=~u4=~u5=~u6=u7=T
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. I=(U,C)=({u1, u2, u3, u4, u5, u6, u7}, {(u1,~u2), (u2,~u3), (~u1,u4), (~u2 ,~u4 ,~u5), (~u4 ,~u7), (u5 , ~u6 ,u7), (u3 , u6)} ~u1=~u2=~u3=~u4=~u5=~u6=u7=T Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

33 Problem of theory of the sets
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Problem of theory of the sets Given a family of sets F, decide whether there exists a family F’, such that: F’ is Helly, Each F’ of F’ has size |F’|>1, For each F of F, U F’ = F F’  F Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

34

35 Our clique graph results
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Our clique graph results 3split3 3split2 s of S has a private neighbor |S| bounded |K| bounded Split graph G = (V,E) partition V=(K,S) ? NPC P |S| < 3 general |K| < 4 Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.

36 Next step If G is a split planar graph => |K| < 4 =>
Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M. Next step If G is a split planar graph => |K| < 4 => => split planar clique is in P. Is CLIQUE NP-complete for planar graphs? Clique graph is NP-complete - Álcon, L., Faria, L., de Figueiredo, C. M. H., Gutierrez, M.