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Partitioning perfect graphs into comparability graphs

András Gyárfás, Márton Marits, Géza Tóth

Abstract

We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.

Partitioning perfect graphs into comparability graphs

Abstract

We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.
Paper Structure (6 sections, 14 theorems, 11 equations, 1 figure)

This paper contains 6 sections, 14 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. The asymptotic of $p(G)$ and $c(G)$
  4. Perfect graph classes with $p(G)\le 2$
  5. Interval graphs
  6. A "small" perfect graph with $c(G)=p(G)=3$

Key Result

Corollary 1

Every perfect graph $G$ can be partitioned into

Figures (1)

  • Figure 1: A graph that is not a comparability graph, represented as the disjointness graph of subtrees of a tree

Theorems & Definitions (18)

  • Corollary 1
  • Theorem 2
  • Definition 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Corollary 8
  • Corollary 9
  • ...and 8 more