Algorithmic Complexity of Secure Connected Domination in Graphs

TL;DR

This paper investigates the computational complexity of secure connected domination in graphs, proving NP-completeness for general graphs and providing efficient algorithms for specific graph classes.

Contribution

It establishes NP-completeness and W[2]-hardness for secure connected domination problems, and offers linear-time solutions for block and threshold graphs.

Findings

NP-complete for split and bipartite graphs

W[2]-hardness results

Linear-time algorithms for block and threshold graphs

Abstract

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected (total) dominating set $S \subseteq V$ is a secure connected (total) dominating set of $G$, if for each $ u \in V \setminus S$, there exists $v \in S$ such that $uv \in E$ and $(S \setminus \lbrace v \rbrace) \cup \lbrace u \rbrace $ 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.