Let \(G=(V,E)\) be a graph. A function \(f : V \rightarrow \{0, 1, 2\}\) is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex \(v \in V\) with \(f (v) = 0\) there is a vertex u adjacent to v with \(f (u) = 2\) and \(\{x\in V:f(x)=0\}\) is an independent set. The weight of f is the value \( f(V)=\sum _{v\in V}f(v)\). An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every \(v\in V\) with \(f(v)>0\) there is a vertex u adjacent to v with \(f (u)>0\). The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by \(\gamma _{oiR}(G)\). Similarly, the outer independent total Roman domination number of G is defined, denoted by \(\gamma _{oitR}(G)\). In this paper, we first show that computing \(\gamma _{oiR}(G)\) (respectively, \(\gamma _{oitR}(G)\)) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph \(G=(V,E)\) we propose an algorithm to compute \(\gamma _{oiR}(G)\) (respectively, \(\gamma _{oitR}(G)\)) in \({\mathcal {O}}(|V| )\) time.

令\（G =（V，E）\）为图。函数\（f：V \ rightarrow \ {0，1，2 \} \）是图形G上的外部独立罗马支配函数（OIRDF），如果每个顶点\（v \ in V \）与\（f （v）= 0 \）与v相邻的顶点u具有\（f（u）= 2 \），并且\（\ {x \ in V：f（x）= 0 \} \）是一个独立集合。f的权重是值\（f（V）= \ sum _ {v \ in V} f（v）\）。外独立总罗马控制函数（OITRDF）˚F上ģ是上OIRDF ģ使得对于每\（V \以V \）与\（F（V）> 0 \）有一个顶点ù相邻v与\（F（U）> 0 \） 。上的OIRDF的最小重量ģ被称为外独立罗马控制数ģ，记\（\伽马_ {OIR}（G）\） 。类似地，定义了G的外部独立总罗马统治数，用\（\ gamma _ {oitR}（G）\）表示。在本文中，我们第一次表明，计算\（\伽马_ {OIR}（G）\） （分别为\（\伽马_ {oitR}（G）\） ）是NP-hard问题，即使当ģ是一个和弦图。然后，对于给定的适当间隔图\（G =（V，E）\） ，我们提出一种算法来计算\（\伽马_ {OIR}（G）\） （分别为\（\伽马_ {oitR}（G）\） ）在\（ {\数学{O}}（| V |）\）时间。