Algorithmic complexity of secure connected domination in graphs

Abstract

Let G ¼ ðV, EÞ be a simple, undirected, and connected graph. A connected (total) dominating set S V is a secure connected (total) dominating set of G, if for each u 2 V n S, there exists v 2 S such that uv 2 E and ðS n fvgÞ [ fug is a connected (total) dominating set of G. The minimum cardinality of a secure connected (total) dominating set of G denoted by c sc ðGÞðc st ðGÞÞ, 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.

Algorithmic aspects of secure connected domination in graphs

“…The secure connected domination number of graph G is the minimum size of a SCDS, and is denoted by γ sc (G) [4]. Given a graph G and a positive integer k, the Secure Connected Domination (SCDM) problem is to check whether G has a SCDS of size at most k. It is known that SCDM is NP-complete for bipartite graphs and split graphs, whereas it is linear time solvable for block graphs and threshold graphs [22]. The Minimum Secure Connected Dominating Set (MSCDS) problem is to find a SCDS of minimum size in the input graph.…”

“…A dominating set S is said to be a connected dominating set (CDS), if the induced subgraph G[S] is connected. A CDS S is said to be a secure connected dominating set (SCDS) in G if for each u ∈ V \ S, there exists v ∈ S such that uv ∈ E and (S \ {v}) ∪ {u} is a CDS in G. Algorithmic complexity of secure connected domination problem has been studied in [10]. In this paper, we initiate the study of a variant of domination called isolate secure domination.…”