Let G be a graph with vertex set V and a subset \(D\subseteq V\). D is a total dominating set of G if every vertex in V is adjacent to a vertex in D. D is a restrained dominating set of G if every vertex in \(V\setminus D\) is adjacent to a vertex in D and another vertex in \(V\setminus D\). D is a total restrained dominating set if D is both a total dominating set and a restrained dominating set. The minimum cardinality of total dominating sets (resp. restrained dominating sets, total restrained dominating sets) of G is called the total domination number (resp. restrained domination number, total restrained domination number) of G, denoted by \(\gamma _{t}(G)\) (resp. \(\gamma _{r}(G)\), \(\gamma _{tr}(G)\)). The MINIMUM TOTAL RESTRAINED DOMINATION (MTRD) problem for a graph G is to find a total restrained dominating set of minimum cardinality of G. The TOTAL RESTRAINED DOMINATION DECISION (TRDD) problem is the decision version of the MTRD problem. In this paper, firstly, we show that the TRDD problem is NP-complete for undirected path graphs, circle graphs, S-CB graphs and C-CB graphs, respectively, and that, for a S-CB graph or C-CB graph with n vertices, the MTRD problem cannot be approximated within a factor of \((1-\epsilon )\textrm{ln} n\) for any \(\epsilon >0\) unless \(NP\subseteq DTIME(n^{O(\textrm{loglog}n)})\). Secondly, for a graph G, we prove that the problem of deciding whether \(\gamma _{r}(G) =\gamma _{tr}(G)\) is NP-hard even when G is restricted to planar graphs with maximum degree at most 4, and that the problem of deciding whether \(\gamma _{t}(G) =\gamma _{tr}(G)\) is NP-hard even when G is restricted to planar bipartite graphs with maximum degree at most 5. Thirdly, we show that the MTRD problem is APX-complete for bipartite graphs with maximum degree at most 4. Finally, we design a linear-time algorithm for solving the MTRD problem for generalized series–parallel graphs.

设G是具有顶点集V和子集\(D\subseteq V\)的图。如果V中的每个顶点都与D中的一个顶点相邻，则D是G的总支配集。如果\(V\setminus D\)中的每个顶点都与D中的一个顶点和\(V\setminus D\)中的另一个顶点相邻，则 D是 G的受约束支配集。如果D既是全支配集又是受约束支配集，则 D 是全约束支配集。G 的总支配集（分别为受约束支配集、总受约束支配集）的最小基数称为G的总支配数（分别为受约束支配数、总受约束支配数），记为\(\gamma _{ t}(G)\)（分别为\(\gamma _{r}(G)\)、\(\gamma _{tr}(G)\) ）。图G的最小总约束支配 (MTRD) 问题是找到G的最小基数的总约束支配集。总约束支配决策 (TRDD) 问题是 MTRD 问题的决策版本。在本文中，首先，我们证明TRDD问题分别对于无向路径图、圆图、S-CB图和C-CB图是NP完全的，并且对于S-CB图或C-CB图对于n 个顶点，对于任何\(\epsilon >0\)， MTRD 问题不能在\((1-\epsilon )\textrm{ln} n\)因子内近似，除非\(NP\subseteq DTIME(n^) {O(\textrm{loglog}n)})\)。其次，对于图G ，我们证明即使G仅限于平面图，判定\(\gamma _{r}(G) =\gamma _{tr}(G)\) 是否是 NP 困难的问题最大度数至多为 4，并且即使G仅限于平面二部图，决定\(\gamma _{t}(G) =\gamma _{tr}(G)\) 是否是 NP 困难的问题最大度数最多为 5。第三，我们证明对于最大度数最多为 4 的二部图，MTRD 问题是 APX 完备的。最后，我们设计了一种线性时间算法来求解广义串并联图的 MTRD 问题。