Computer Science > Data Structures and Algorithms
[Submitted on 11 Oct 2023 (v1), last revised 24 Jan 2024 (this version, v2)]
Title:Algorithmic study on liar's vertex-edge domination problem
View PDF HTML (experimental)Abstract:Let $G=(V,E)$ be a graph. For an edge $e=xy\in E$, the closed neighbourhood of $e$, denoted by $N_G[e]$ or $N_G[xy]$, is the set $N_G[x]\cup N_G[y]$. A vertex set $L\subseteq V$ is liar's vertex-edge dominating set of a graph $G=(V,E)$ if for every $e_i\in E$, $|N_G[e_i]\cap L|\geq 2$ and for every pair of distinct edges $e_i$ and $e_j$, $|(N_G[e_i]\cup N_G[e_j])\cap L|\geq 3$. This paper introduces the notion of liar's vertex-edge domination which arises naturally from some applications in communication networks. Given a graph $G$, the \textsc{Minimum Liar's Vertex-Edge Domination Problem} (\textsc{MinLVEDP}) asks to find a liar's vertex-edge dominating set of $G$ of minimum cardinality. In this paper, we study this problem from algorithmic point of view. We show that \textsc{MinLVEDP} can be solved in linear time for trees, whereas the decision version of this problem is NP-complete for chordal graphs, bipartite graphs, and $p$-claw free graphs for $p\geq 4$. We further study approximation algorithms for this problem. We propose two approximation algorithms for \textsc{MinLVEDP} in general graphs and $p$-claw free graphs.
%We propose an $O(\ln \Delta(G))$-approximation algorithm for \textsc{MinLVEDP} in general graphs, where $\Delta(G)$ is the maximum degree of the input graph. Also, we design a constant factor approximation algorithm for $p$-claw free graphs.
On the negative side, we show that the \textsc{MinLVEDP} cannot be approximated within $\frac{1}{2}(\frac{1}{8}-\epsilon)\ln|V|$ for any $\epsilon >0$, unless $NP\subseteq DTIME(|V|^{O(\log(\log|V|)})$. Finally, we prove that the \textsc{MinLVEDP} is APX-complete for bounded degree graphs and $p$-claw free graphs for $p\geq 6$.
Submission history
From: Subhabrata Paul [view email][v1] Wed, 11 Oct 2023 13:07:59 UTC (620 KB)
[v2] Wed, 24 Jan 2024 13:08:53 UTC (27 KB)
Current browse context:
cs.DS
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.