Paired Domination versus Domination and Packing Number in Graphs
Magda Dettlaff, Didem G\"oz\"upek, Joanna Raczek

TL;DR
This paper investigates the computational complexity of relationships between various domination and packing parameters in graphs, proving NP-hardness results and providing characterizations and algorithms for specific cases.
Contribution
It establishes NP-hardness of determining equality between paired domination and other parameters, and characterizes trees with a specific equality, offering new insights and algorithms.
Findings
NP-hard to decide if γ_pr(G) = 2γ(G) for bipartite graphs
Polynomial-time recognition of trees with γ_pr(G) = 2γ(G)
NP-hard to determine if γ_pr(G) = γ_t(G) and related equalities
Abstract
Given a graph , the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph are denoted by , , and , respectively. For a positive integer , a -packing in is a set such that for every pair of distinct vertices and in , the distance between and is at least . The -packing number is the order of a largest -packing and is denoted by . It is well known that . In this paper, we prove that it is NP-hard to determine whether even for bipartite graphs. We provide a simple characterization of trees with , implying a polynomial-time recognition algorithm. We also prove that even for a bipartite graph, it is…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Interconnection Networks and Systems · Graph Labeling and Dimension Problems
