For a simple, undirected graph G, a function h : V (G) →{0, 1, 2} which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of G with weight h(V ) =∑v∈Vh(v). (C1) For all q ∈ V with h(q) = 0 there exists a vertex r such that qr ∈ E and h(r) = 2, (C2) The induced subgraph with vertex set {p : h(p) ≥ 1} has no isolated vertices and (C3) The induced subgraph with vertex set {p : h(p) = 0} is independent. For a graph G, the smallest possible weight of an OITRDF of G which is denoted by γoitR(G), is known as the outer-independent total Roman domination number of G. The problem of determining γoitR(G) of a graph G is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if G has an OITRDF of weight at most l for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

中文翻译：

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图中外部独立的完全罗马统治的算法方面

对于一个简单的无向图G, 一个函数H ： 五 (G) →{0, 1, 2}满足以下条件的称为外部独立全罗马支配函数（OITRDF）G有重量H(五 ) =∑v∈五H(v). (C1) 为所有人q ∈ 五和H(q) = 0存在一个顶点r这样qr ∈ 乙和H(r) = 2, (C2) 带顶点集的诱导子图{p ： H(p) ≥ 1}没有孤立的顶点和 (C3) 具有顶点集的诱导子图{p ： H(p) = 0}是独立的。对于图表G, OITRDF 的最小可能权重为G这表示为γ○一世吨R(G), 被称为外部独立的总罗马统治数的G. 确定问题γ○一世吨R(G)图的G被称为最小外部独立总罗马统治问题（MOITRDP）。在本文中，我们展示了决定是否G至多有一个重量为 OITRDFl对于二部图和分裂图，弦图的子类是 NP 完全的。我们还表明 MOITRDP 对于连接的阈值图和有界树宽图是线性时间可解的。最后，我们表明支配和外部独立的总罗马支配问题在计算复杂性方面并不等价。