Let G(V, E) be a simple, connected and undirected graph. A dominating set \(S \subseteq V\) is called a 2-secure dominating set (2-SDS) in G, if for each pair of distinct vertices \(v_1,v_2 \in V\) there exists a pair of distinct vertices \(u_1,u_2 \in S\) such that \(u_1 \in N[v_1]\), \(u_2 \in N[v_2]\) and \((S {\setminus } \{u_1,u_2\}) \cup \{v_1,v_2 \}\) is a dominating set in G. The size of a minimum 2-SDS in G is said to be 2-secure domination number denoted by \(\gamma _{2s}(G)\). The 2-SDM problem is to check if an input graph G has a 2-SDS S, with \( \vert S \vert \le k\), where \( k \in \mathbb {Z}^+ \). It is proved that for bipartite graphs 2-SDM is NP-complete. In this paper, we prove that the 2-SDM problem is NP-complete for planar graphs and doubly chordal graphs, a subclass of chordal graphs. We reinforce the existing NP-complete result for bipartite graphs, by proving 2-SDM is NP-complete for some subclasses of bipartite graphs specifically, comb convex bipartite and star convex bipartite graphs. We prove that this problem is linear time solvable for bounded tree-width graphs. We also show that the 2-SDM is W[2]-hard even for split graphs. The M2SDS problem is to find a 2-SDS of minimum size in the given graph. We give a \( \varDelta +1 \)-approximation algorithm for M2SDS, where \( \varDelta \) is the maximum degree of the given graph and prove that M2SDS cannot be approximated within \( (1 - \epsilon ) \ln (\vert V \vert ) \) for any \( \epsilon > 0 \) unless \( NP \subseteq DTIME(\vert V \vert ^{ O(\log \log \vert V \vert )}) \). Finally, we prove that the M2SDS is APX-complete for graphs with \(\varDelta =4.\)

令G（V，  E）是一个简单的，连通的，无向的图。如果对每对不同的顶点\（v_1，v_2 \ in V \）中存在一对不同的对，则控制集\（S \ subseteq V \）在G中称为2-安全控制集（2-SDS）。顶点\（u_1，u_2 \ in S \）使得\（u_1 \ in N [v_1] \），\（u_2 \ in N [v_2] \）和\（（S {\ setminus} \ {u_1，u_2 \}）\ cup \ {v_1，v_2 \} \）是G中的主要组。G中最小的2-SDS的大小被称为2-安全控制数，表示为\（\ gamma _ {2s}（G）\）。2-SDM问题是检查输入图G是否具有2-SDS S，其中\（\ vert S \ vert \ le k \），其中\（k \ in \ mathbb {Z} ^ + \）。证明对于二部图，2-SDM是NP完全的。在本文中，我们证明了平面图和双弦和弦图（弦和图的子类）的2-SDM问题是NP完全的。通过证明2-SDM对于二分图的某些子类，尤其是梳凸二分图和星形凸二分图，我们加强了二分图的现有NP完全结果。我们证明该问题对于有界树宽图是线性时间可解决的。我们还显示，即使对于拆分图，2-SDM也是W [2]硬的。M2SDS问题是在给定图中找到最小尺寸的2-SDS。我们为M2SDS给出了\（\ varDelta +1 \） -近似算法，其中\（\ varDelta \）是给定的图形的最大程度，并证明M2SDS不能内近似\（（1 - \小量）\ LN（\ VERT V \ VERT）\）对于任何\（\小量> 0 \）除非\（NP \子集DTIME（\ vert V \ vert ^ {O（\ log \ log \ vert V \ vert}}）\）。最后，我们证明对于\（\ varDelta = 4。\）的图，M2SDS是APX完全的。