Split clique graph complexity L. Alcn and M. Gutierrez La Plata, - PowerPoint PPT Presentation

Split clique graph complexity L. Alcón and M. Gutierrez La Plata, Argentina L. Faria and C. M. H. de Figueiredo, Rio de Janeiro, Brazil 1

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 2 3

Our Problem 2 1 2 1 4 3 H G 4 3 3

Previous result • WG’2006 – CLIQUE GRAPH is NPC for graphs with maximum degree 14 and maximum clique size 12 4

Classes of graphs where CLIQUE GRAPH is polynomial 2 1 2 1 4 3 4 G H 3 Survey of Jayme L. Szwarcfiter 5

A quest for a non-trivial Polynomial decidable class 6

Chordal? • Clique structure, simplicial elimination sequence, … ? Split? • Same as chordal, one clique and one independent set … ? 7

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 8

Main used statements 9

ksplit p • Given a pair of integers k > p, G is ksplit p if G is a split (K,S) graph and for every vertex s of S, p < d(s) < k. Example of 3split 2 10

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 3split 2 11

1. |S| < 3 s 1 N(s 1 ) s 2 s 3 N(s 2 ) N(s 3 ) G is clique graph iff G is not 12

2. |K| < 4 G is clique graph iff G has no bases set with 13

3. s has a private neighbour • G is a clique graph 14

Our NPC problem CLIQUE GRAPH 3split 2 NPC NPC INSTANCE : A 3 3split 2 graph G=(V,E) with partition (K,S) QUESTION : Is there a graph H such that G=K(H)? 3SAT 3 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? 15

Some preliminaries • Black vertices in K • White vertices in S • Theorem – K is assumed in every 3split 2 RS-family 16

3split 2 17

3split 2 18

The evil triangle s 1 s 2 k 1 k 3 k 2 s 3 19

The main gadget Variable 20

21

22

23

24

25

26

Variable 27

28

29

Clause 30

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)} 31

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)} 32 ~u1=~u2=~u3=~u4=~u5=~u6=u7=T

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 33

34

Our clique graph results s of S has a 3split3 3split2 |S| bounded |K| bounded private neighbor Split graph |S| < 3 general |K| < 4 general G = (V,E) ? NPC P partition V=(K,S) P ? P ? 35

Next step • If G is a split planar graph => |K| < 4 => => split planar clique is in P. • Is CLIQUE NP-complete for planar graphs? 36