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On the $d$-independence number in 1-planar graphs

Therese Biedl, Prosenjit Bose, Babak Miraftab

Abstract

The $d$-independence number of a graph $G$ is the largest possible size of an independent set $I$ in $G$ where each vertex of $I$ has degree at least $d$ in $G$. Upper bounds for the $d$-independence number in planar graphs are well-known for $d=3,4,5$, and can in fact be matched with constructions that actually have minimum degree $d$. In this paper, we explore the same questions for 1-planar graphs, i.e., graphs that can be drawn in the plane with at most one crossing per edge. We give upper bounds for the $d$-independence number for all $d$. Then we give constructions that match the upper bound, and (for small $d$) also have minimum degree $d$.

On the $d$-independence number in 1-planar graphs

Abstract

The -independence number of a graph is the largest possible size of an independent set in where each vertex of has degree at least in . Upper bounds for the -independence number in planar graphs are well-known for , and can in fact be matched with constructions that actually have minimum degree . In this paper, we explore the same questions for 1-planar graphs, i.e., graphs that can be drawn in the plane with at most one crossing per edge. We give upper bounds for the -independence number for all . Then we give constructions that match the upper bound, and (for small ) also have minimum degree .

Paper Structure

This paper contains 18 sections, 17 theorems, 9 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper bounds on the d-independence number
  4. Upper bounds on the d-independence number for d=3, 4, 5
  5. Upper bounds for on the 6-independence number
  6. Upper bounds for d ≥ 7
  7. 1-planar graphs with large d-independent sets
  8. Constructions that match Lemma 4.2
  9. The case $d=3$:
  10. The standard construction:
  11. The case $d=4,5$:
  12. Planar pairings:
  13. Simple graphs
  14. Exploiting planar pairings and claw-covers:
  15. Filling Table \ref{['ta:resultsNew']}:
  16. ...and 3 more sections

Key Result

Lemma 1

BiedlW21 Let $G$ be a simple $1$-planar graph. Let $I$ be a non-empty independent set in $G$ where $\deg(t)\geq 3$ for all $t\in I$. Let $I_d$ be the vertices in $I$ that have degree $d$. Then

Figures (11)

  • Figure 1: The graph $H_3$ (drawn in two different ways) and the graph $G_3$ with a 3-independent set of size $n-2$. Dotted lines indicate the planar pairing.
  • Figure 2: The standard construction on the standing cylinder and in the plane.
  • Figure 3: The graphs $G_d$ for $d=4$ and $d=5$ (construction shown for $s=3$).
  • Figure 4: Graphs $G_6$ and $G_7$ are created by taking $G_3$ and $G_4$, respectively, and inserting $H_3$ (light blue) at each pair of the planar pairing.
  • Figure 5: Simple graphs for the $d$-independence number, for $d=3$ ($s=3$), $d=6$ ($s=4$) and $d=7$ ($s=5$).
  • ...and 6 more figures

Theorems & Definitions (39)

  • Lemma 1
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Lemma 2
  • proof
  • Corollary 4
  • ...and 29 more