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Graphs with unique zero forcing sets and Grundy dominating sets
arXiv - CS - Discrete Mathematics Pub Date : 2021-03-18 , DOI: arxiv-2103.10172
Boštjan Brešar, Tanja Dravec

The concept of zero forcing was introduced in the context of linear algebra, and was further studied by both graph theorists and linear algebraists. It is based on the process of activating vertices of a graph $G$ starting from a set of vertices that are already active, and applying the rule that an active vertex with exactly one non-active neighbor forces that neighbor to become active. A set $S\subset V(G)$ is called a zero forcing set of $G$ if initially only vertices of $S$ are active and the described process enforces all vertices of $G$ to become active. The size of a minimum zero forcing set in $G$ is called the zero forcing number of $G$. While a minimum zero forcing set can only be unique in edgeless graphs, we consider the weaker uniqueness condition, notably that for every two minimum zero forcing sets in a graph $G$ there is an automorphism that maps one to the other. We characterize the class of trees that enjoy this condition by using properties of minimum path covers of trees. In addition, we investigate both variations of uniqueness for several concepts of Grundy domination, which first appeared in the context of domination games, yet they are also closely related to zero forcing. For each of the four variations of Grundy domination we characterize the graphs that have only one Grundy dominating set of the given type, and characterize those forests that enjoy the weaker (isomorphism based) condition of uniqueness. The latter characterizations lead to efficient algorithms for recognizing the corresponding classes of forests.

中文翻译:

具有唯一的零强迫集和Grundy支配集的图

零强迫的概念是在线性代数的背景下引入的,并且由图形理论家和线性代数论者进行了进一步的研究。它基于以下过程:从一组已经处于活动状态的顶点开始激活图$ G $的顶点,并应用以下规则:具有正好一个非活动邻居的活动顶点会强制该邻居变为活动状态。如果最初只有$ S $的顶点处于活动状态,并且所描述的过程强制将$ G $的所有顶点变为活动状态,则将一组$ S \子集V(G)$称为$ G $的归零集合。$ G $中设置的最小零强制大小的大小称为$ G $的零强制数目。虽然最小的零强迫集只能在无边图中唯一,但我们认为唯一性条件较弱,值得注意的是,在图形$ G $中,每两个最小的零强迫集都有一个自同构性,将一个映射到另一个。我们通过使用树木的最小路径覆盖的属性来表征享受这种条件的树木的类别。此外,我们研究了Grundy统治的几个概念的唯一性的两种变化,这些概念首先出现在统治游戏的背景下,但它们也与零强迫密切相关。对于Grundy支配的四个变体中的每一个,我们表征仅具有给定类型的一个Grundy支配集的图,并表征那些具有较弱(基于同构)唯一性条件的森林。后者的特征导致了用于识别森林的相应类别的有效算法。我们通过使用树木的最小路径覆盖的属性来表征享受这种条件的树木的类别。此外,我们研究了Grundy统治的几个概念的唯一性的两种变化,这些概念首先出现在统治游戏的背景下,但它们也与零强迫密切相关。对于Grundy支配的四个变体中的每一个,我们表征仅具有给定类型的一个Grundy支配集的图,并表征那些具有较弱(基于同构性)唯一性条件的森林。后者的特征导致了用于识别森林的相应类别的有效算法。我们通过使用树木的最小路径覆盖的属性来表征享受这种条件的树木的类别。此外,我们研究了Grundy统治的几个概念的唯一性的两种变化,这些概念首先出现在统治游戏的背景下,但它们也与零强迫密切相关。对于Grundy支配的四个变体中的每一个,我们表征仅具有给定类型的一个Grundy支配集的图,并表征那些具有较弱(基于同构)唯一性条件的森林。后者的特征导致了用于识别森林的相应类别的有效算法。它首先出现在统治游戏的背景下,但它们也与零强迫密切相关。对于Grundy支配的四个变体中的每一个,我们表征仅具有给定类型的一个Grundy支配集的图,并表征那些具有较弱(基于同构性)唯一性条件的森林。后者的特征导致了用于识别森林的相应类别的有效算法。它首先出现在统治游戏的背景下,但它们也与零强迫密切相关。对于Grundy支配的四个变体中的每一个,我们表征仅具有给定类型的一个Grundy支配集的图,并表征那些具有较弱(基于同构)唯一性条件的森林。后者的特征导致了用于识别森林的相应类别的有效算法。
更新日期:2021-03-19
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