关于图中若干分离-支配码基数的下界
On lower bounds for cardinalities of several separating-dominating codes in graphs
AI总结：
本文研究图中分离-支配码基数的下界问题，针对八个识别问题提供极图的一般构造，重新证明并建立了对数下界，同时刻画了达到这些下界的所有图。
摘要：
文献中研究了图中的几个不同识别问题，其中研究最广泛的是基于支配集作为识别工具的问题。其目标是通过在适当选取的支配集或全支配集中的唯一邻域来分离图中的任意两个顶点。这种带有分离性质的（全）支配集通常被称为图的“码”。确定此类最小基数码的问题均被证明是 NP-hard 的。因此，解决此类问题的典型思路是为所研究的码的基数提供界限。本文关注关于图阶的下界的极图。对于某些码，文献中已知对数下界。针对八个不同的识别问题，我们提供了一种极图的一般构造，使得与图阶相比，最小码的基数达到最小。这使我们能够重新证明现有的对数下界，为其他码建立此类界限，并刻画达到这些界限的所有图。
英文摘要：
In the literature, several different identification problems in graphs have been studied, the most widely studied such problems are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a "code" of the graph. The problems of determining such codes of minimum cardinality are all shown to be NP-hard. A typical line to attack such problems is, therefore, to provide bounds on the cardinalities of the studied codes. In this paper, we are interested in extremal graphs for lower bounds in terms of the order of the graph. For some codes, logarithmic lower bounds are known from the literature. We provide for eight different identification problems a general construction of extremal graphs minimizing the cardinality of a minimum code compared to the order of the graph. This enables us to reprove the existing logarithmic lower bounds, to establish such bounds for further codes, and to characterize all graphs attaining these bounds.
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