In this paper, we study Roman {k}-dominating functions on a graph G with vertex set V for a positive integer k: a variant of {k}-dominating functions, generations of Roman \(\{2\}\)-dominating functions and the characteristic functions of dominating sets, respectively, which unify classic domination parameters with certain Roman domination parameters on G. Let \(k\ge 1\) be an integer, and a function \(f:V \rightarrow \{0,1,\dots ,k\}\) defined on V called a Roman \(\{k\}\)-dominating function if for every vertex \(v\in V\) with \(f(v)=0\), \(\sum _{u\in N(v)}f(u)\ge k\), where N(v) is the open neighborhood of v in G. The minimum value \(\sum _{u\in V}f(u)\) for a Roman \(\{k\}\)-dominating function f on G is called the Roman \(\{k\}\)-domination number of G, denoted by \(\gamma _{\{Rk\}}(G)\). We first present bounds on \(\gamma _{\{Rk\}}(G)\) in terms of other domination parameters, including \(\gamma _{\{Rk\}}(G)\le k\gamma (G)\). Secondly, we show one of our main results: characterizing the trees achieving equality in the bound mentioned above, which generalizes M.A. Henning and W.F. klostermeyer’s results on the Roman {2}-domination number (Henning and Klostermeyer in Discrete Appl Math 217:557–564, 2017). Finally, we show that for every fixed \(k\in \mathbb {Z_{+}}\), associated decision problem for the Roman \(\{k\}\)-domination is NP-complete, even for bipartite planar graphs, chordal bipartite graphs and undirected path graphs.

在本文中，我们研究在图G上具有正整数k的顶点集V的罗马{ k }支配函数：{ k }支配函数的变体，是罗马\（\ {2 \} \）的代-支配函数和支配集的特征函数，分别将经典支配参数与G上的某些罗马支配参数统一起来。令\（k \ ge 1 \）为整数，在V上定义的函数\（f：V \ rightarrow \ {0,1，\ dots，k \} \）为罗马\（\ {k \} \） -如果每个顶点\（v \ in V \）具有\（f（v）= 0 \），\（\ sum _ {u \ in N（v）} f（u）\ ge k \），其中N（v）是v在G中的开放邻域。G上以罗马\（\ {k \} \）为主导函数f的最小值\（\ sum _ {u \ in V} f（u）\）称为罗马\（\ {k \} \ ） -domination数的ģ，记\（\伽马_ {\ {RK \}}（G）\） 。我们首先根据其他控制参数，包括\（\ gamma _ {\ {Rk \}}（G）\ le k \ gamma，给出\（\ gamma _ {\ {Rk \}}（G）\）的边界（G）\）。其次，我们展示了我们的主要结果之一：表征在上述范围内达到平等的树木，这将MA Henning和WF klostermeyer的结果推广到罗马{2}支配数上（Henning和Klostermeyer在“离散应用数学217：557”中564，2017）。最后，我们表明，对于每个固定的\（k \ in \ mathbb {Z _ {+}} \），罗马\（\ {k \} \）-支配的相关决策问题都是NP完全的，即使对于二分平面图，弦二部图和无向路径图。