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

TL;DR

This paper investigates the $d$-independence number in 1-planar graphs, providing upper bounds and constructions that match these bounds, extending known results from planar graphs to 1-planar graphs.

Contribution

It establishes upper bounds for the $d$-independence number in 1-planar graphs and constructs examples that meet these bounds, including minimum degree conditions for small $d$.

Findings

Upper bounds for $d$-independence number in 1-planar graphs for all $d$

Constructed graphs matching the upper bounds

Results extend planar graph bounds to 1-planar graphs

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 $d$.