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 $d$.

翻译：图$G$的$d$-独立数是指$G$中满足以下条件的最大独立集$I$的规模：$I$中的每个顶点在$G$中的度至少为$d$。对于平面图，当$d=3,4,5$时，$d$-独立数的上界已有明确结论，且这些上界可通过实际最小度为$d$的构造达到。本文针对1-平面图（即每条边在平面绘制中至多有一个交叉的图）探讨相同问题。我们给出了所有$d$对应的$d$-独立数上界，并构造出达到该上界的图例，同时（对于较小的$d$）这些构造图还具有最小度$d$的特性。