The problems of determining the minimum-sized \emph{identifying}, \emph{locating-dominating} and \emph{open locating-dominating codes} of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set $C$ of a graph $G$ such that the vertices of a chosen subset of $V(G)$ (i.e. either $V(G)\setminus C$ or $V(G)$ itself) are uniquely determined by their neighborhoods in $C$. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graphs classes. In this work, we present tight lower and upper bounds for all three types of codes for \emph{block graphs} (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or \emph{blocks}) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature - with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

翻译：确定输入图的最小规模\emph{识别码}、\emph{定位支配码}与\emph{开放定位支配码}问题是特殊的搜索问题，在理论与计算层面均具有挑战性。在这些问题中，需选取图$G$的支配集$C$，使得$V(G)$的选定子集（即$V(G)\setminus C$或$V(G)$本身）中的顶点能通过其在$C$中的邻域唯一确定。解决此类问题的典型思路是确定各类图结构中最小码的紧界。本研究针对\emph{块图}（即无钻石弦图）的所有三类编码给出了紧的上下界。我们的界以块图的最大团（或称\emph{块}）数量及图的阶数为参数。其中两个上界验证了文献中的猜想——本文证明了其中一个猜想在块图上的成立性。对于下界，我们证明其关于块数及块图阶数均呈线性关系。通过构造达到这些界的最小码块图族，我们证明了每个界的紧性。