A sequence of vertices \((v_1,\, \dots , \,v_k)\) of a graph G is called a dominating closed neighborhood sequence if \(\{v_1,\, \dots , \,v_k\}\) is a dominating set of G and \(N[v_i]\nsubseteq \cup _{j=1}^{i-1} N[v_j]\) for every i. A graph G is said to be \(k-\)uniform if all dominating closed neighborhood sequences in the graph have equal length k. Brešar et al. (Discrete Math 336:22–36, 2014) characterized k-uniform graphs with \(k\le 3\). In this article we extend their work by giving a complete characterization of all k-uniform graphs with \(k\ge 4\).

中文翻译：

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均匀长度支配序列图

顶点的序列\（（V_1，\，\点，\，V_K）\）的曲线图的ģ称为主导闭邻域序列如果\（\ {V_1，\，\点，\，V_K \} \）是每个i的G和\（N [v_i] \ nsubseteq \ cup _ {j = 1} ^ {i-1} N [v_j] \）的主导集合。如果图中的所有主导闭合邻域序列都具有相等的长度k，则图G被称为\（k- \）均匀。Brešar等。（离散数学336：22–36，2014年）用\（k \ le 3 \）表征k个均匀图。在本文中，我们通过使用\（k \ ge 4 \）给出所有k均匀图的完整特征来扩展其工作。