图的自同态破缺
Endomorphism Breaking in Graphs
摘要：
我们引入图$G$的{ \it自同态可区别数} $D_e(G)$作为最小基数$d$，使得$G$有一个顶点着色为仅由平凡自同态保持的$d$颜色.这推广了图$G$的区别数$D(G)$的概念，它是为自同构而不是自同态定义的。由于自同态的数量可以大大超过自同构的数量，新概念带来了挑战性的问题，其中几个在这里提出。特别是，我们调查之间的关系$D_e(G)$和自同态运动的图$G$，即，最少可能的数量移动的顶点的非平凡自同态的$G$。此外，我们还将关于有限图和无限图的可区别数的许多结果推广到自同态可区别数。这是本文主要关注的问题。
英文摘要：
We introduce the {\it endomorphism distinguishing number} $D_e(G)$ of a graph $G$ as the least cardinal $d$ such that $G$ has a vertex coloring with $d$ colors that is only preserved by the trivial endomorphism. This generalizes the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined for automorphisms instead of endomorphisms. As the number of endomorphisms can vastly exceed the number of automorphisms, the new concept opens challenging problems, several of which are presented here. In particular, we investigate relationships between $D_e(G)$ and the endomorphism motion of a graph $G$, that is, the least possible number of vertices moved by a nontrivial endomorphism of $G$. Moreover, we extend numerous results about the distinguishing number of finite and infinite graphs to the endomorphism distinguishing number. This is the main concern of the paper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
