Let G be some simple graph and k be any positive integer. Take \(h: V(G)\rightarrow \{0,1,\ldots ,k+1\}\) and \(v \in V(G)\), let \(AN_{h}(v)\) denote the set of vertices \(w\in N_{G}(v)\) with \(h(w)\ge 1\). Let \(AN_{h}[v] = AN_{h}(v)\cup \{v\}\). The function h is a [k]-Roman dominating function of G if \(h(AN_{h}[v]) \ge |AN_{h}(v)| + k\) holds for any \(v \in V(G)\). The minimum weight of such a function is called the k-th Roman Domination number of G, which is denoted by \(\gamma _{kR}(G)\). In 2020, Banerjee et al. presented linear time algorithms to compute the double Roman domination number on proper interval graphs and block graphs. They posed the open question that whether there is some polynomial time algorithm to solve the double Roman domination problem on interval graphs. It is argued that the interval graph is a nontrivial graph class. In this article, we design a simple dynamic polynomial time algorithm to solve the k-th Roman domination problem on interval graphs for each fixed integer \(k>1\).

令G为某个简单图， k为任意正整数。取\(h: V(G)\rightarrow \{0,1,\ldots ,k+1\}\)和\(v \in V(G)\) ，令\(AN_{h}(v) \)表示顶点集\(w\in N_{G}(v)\)和\(h(w)\ge 1\) 。令\(AN_{h}[v] = AN_{h}(v)\cup \{v\}\) 。函数h是G的 [ k ]-罗马支配函数，如果\(h(AN_{h}[v]) \ge |AN_{h}(v)| + k\)对于任何\(v \in V(G)\) 。这种函数的最小权重称为G的第 k 个罗马统治数，用\(\gamma _{kR}(G)\)表示。 2020 年，Banerjee 等人。提出了线性时间算法来计算适当区间图和块图上的双罗马支配数。他们提出了一个悬而未决的问题：是否存在某种多项式时间算法来解决区间图上的双罗马支配问题。有人认为区间图是一个非平凡的图类。在本文中，我们设计了一个简单的动态多项式时间算法来解决区间图上每个固定整数\(k>1\)的第 k 个罗马支配问题。