Abstract
This paper is motivated by the concept of the signed k-domination problem and dedicated to the complexity of the problem on graphs. For any fixed nonnegative integer k, we show that the signed k-domination problem is NP-complete for doubly chordal graphs. For strongly chordal graphs and distance-hereditary graphs, we show that the signed k-domination problem can be solved in polynomial time. We also show that the problem is linear-time solvable for trees, interval graphs, and chordal comparability graphs.
Highlights
Let G = (V, E) be a finite, undirected, simple graph
We study the signed k-domination problem for several well-known classes of graphs such as doubly chordal graphs, strongly chordal graphs, distance-hereditary graphs, trees, interval graphs, and chordal comparability graphs
We show the NP-completeness of the signed k-domination problem on doubly chordal graphs by a polynomial-time reduction from the signed (k −1) -domination problem on doubly chordal graphs
Summary
In 2012, Wang [1] studied the notion of signed k-domination on graphs as follows. Before presenting the NP-complete results, we restate the signed k-domination problem as decision problems as follows: Given a graph G = (V, E) and a nonnegative integer k and an integer λ , is γ k,S (G) ≤ λ ? Theorem 1 [3] [4] For any integer k = 0 or 1, the signed k-domination problem on doubly chordal graphs and bipartite planar graphs is NP-complete. For any fixed integer k ≥ 2, the signed k-domination problem on doubly chordal graphs is NP-complete. The signed k-domination problem on doubly chordal graphs is in NP. 0-domination and 1-domination problems on doubly chordal graphs are NP-complete. It implies that for any integer λ, γ k−1,S (G) ≤ λ if and only if γ k,S (H ) ≤ λ + n ⋅ k − n +1
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
