Algorithmic complexity of secure connected domination in graphs

Let G=(V,E) be a simple, undirected, and connected graph. A connected (total) dominating set S subset of V is a secure connected (total) dominating set of G, if for each u is an element of V\S, there exists v is an element of S such that uv is an element of E and (S\{v}){u} is a connected (total) dominating set of G. The minimum cardinality of a secure connected (total) dominating set of G denoted by gamma sc(G)(gamma st(G)), is called the secure connected (total) domination number of G. In this paper, we show that the decision problems corresponding to secure connected domination number and secure total domination number are NP-complete even when restricted to split graphs or bipartite graphs. The NP-complete reductions also show that these problems are w[2]-hard. We also prove that the secure connected domination problem is linear time solvable in block graphs and threshold graphs.