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

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

Aug 04, 2023 201 likes •574 views

Split clique graph complexity L. Alcn 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

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Split clique graph complexity

  • L. Alcón and M. Gutierrez

La Plata, Argentina

  • L. Faria and C. M. H. de Figueiredo,

Rio de Janeiro, Brazil

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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. 1 2 3 4 1 2 3 4

H G

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Our Problem

1 2 3 4

H

1 2 3 4

G

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Previous result

  • WG’2006 – CLIQUE GRAPH is NPC for

graphs with maximum degree 14 and maximum clique size 12

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Classes of graphs where CLIQUE GRAPH is polynomial

1 2 3 4

H

1 2 3 4

G

Survey of Jayme L. Szwarcfiter

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A quest for a non-trivial

Polynomial decidable class

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Chordal?

  • Clique structure, simplicial elimination sequence, … ?

Split?

  • Same as chordal, one clique and one

independent set … ?

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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

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Main used statements

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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

  • f 3split2
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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
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  • 1. |S| < 3

G is clique graph iff G is not N(s2) N(s1) N(s3) s1 s3 s2

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  • 2. |K| < 4

G is clique graph iff G has no bases set with

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

neighbour

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Our NPC problem NPC NPC

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)?

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

  • ccurs once, each negative literal occurs once or twice.

QUESTION: Is there a truth assignment for U satisfying each clause of C?

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Some preliminaries

  • Black vertices in K
  • White vertices in S
  • Theorem – K is assumed in every 3split2

RS-family

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3split2

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3split2

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The evil triangle

s1 s2 k1 k3 k2 s3

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The main gadget

Variable

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Variable

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Clause

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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)}

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32 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

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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

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35 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 general P ? P ?

Our clique graph results

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Next step

  • Is CLIQUE NP-complete for planar graphs?
  • If G is a split planar graph => |K| < 4 =>

=> split planar clique is in P.