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Disjoint connected dominating sets in pseudorandom graphs

Nemanja Draganić, Michael Krivelevich

Abstract

A connected dominating set (CDS) in a graph is a dominating set of vertices that induces a connected subgraph. Having many disjoint CDSs in a graph can be considered as a measure of its connectivity, and has various graph-theoretic and algorithmic implications. We show that $d$-regular (weakly) pseudoreandom graphs contain $(1+o(1))d/\ln d$ disjoint CDSs, which is asymptotically best possible. In particular, this implies that random $d$-regular graphs typically contain $(1+o(1))d/\ln d$ disjoint CDSs.

Disjoint connected dominating sets in pseudorandom graphs

Abstract

A connected dominating set (CDS) in a graph is a dominating set of vertices that induces a connected subgraph. Having many disjoint CDSs in a graph can be considered as a measure of its connectivity, and has various graph-theoretic and algorithmic implications. We show that -regular (weakly) pseudoreandom graphs contain disjoint CDSs, which is asymptotically best possible. In particular, this implies that random -regular graphs typically contain disjoint CDSs.

Paper Structure

This paper contains 6 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of $(n,d,\lambda)$-graphs
  4. The Friedman-Pippenger tree embedding technique with rollbacks
  5. Proof of \ref{['thm:main']}
  6. Concluding remarks

Key Result

Theorem 2

For every $\varepsilon>0$, there exists a $C=C(\varepsilon)$ so that the following holds. Every $(n,d,\lambda)$-graph with $d/\lambda>C$ contains at least $(1-\varepsilon)\frac{d}{\ln d}$ disjoint connected dominating sets.

Theorems & Definitions (15)

  • Definition 1
  • Theorem 2
  • Lemma 3: Expander Mixing Lemma
  • Lemma 4
  • proof
  • Remark 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 9: Corollary 3.7 in montgomery2019spanning
  • ...and 5 more