Liar's vertex-edge domination in subclasses of chordal graphs

2025-07-07

AI Summary

This paper investigates the Minimum Liar Vertex–Edge Dominating Set problem (MinLVEDP) on subclasses of chordal graphs—specifically block graphs and proper interval graphs. MinLVEDP seeks a minimum vertex set (L) such that (i) the closed neighborhood of every edge intersects (L) in at least two vertices, and (ii) the union of the closed neighborhoods of any two distinct edges intersects (L) in at least three vertices. We introduce this model for the first time and establish a unified algorithmic framework. For block graphs and proper interval graphs, we design linear-time dynamic programming algorithms leveraging structural properties of edge closed neighborhoods and graph decomposition. In contrast, we prove that MinLVEDP is NP-complete on undirected path graphs, thereby establishing its computational complexity boundary. Our results provide both theoretical foundations and efficient, deployable algorithms for fault-tolerant coverage in communication networks.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Let $G=(V, E)$ be an undirected graph. The set $N_G[x]={yin V|xyin E}cup {x}$ is called the closed neighbourhood of a vertex $xin V$ and for an edge $e=xyin E$, the closed neighbourhood of $e$ is the set $N_G[x]cup N_G[y]$, which is denoted by $N_G[e]$ or $N_G[xy]$. A set $Lsubseteq V$ is called emph{liar's vertex-edge dominating set} of a graph $G=(V,E)$ if for every $e_iin E$, $|N_G[e_i]cap L|geq 2$ and for every pair of distinct edges $e_i,e_jin E$, $|(N_G[e_i]cup N_G[e_j])cap L|geq 3$. The notion of liar's vertex-edge domination arises naturally from some applications in communication networks. Given a graph $G$, the extsc{Minimum Liar's Vertex-Edge Domination Problem} ( extsc{MinLVEDP}) asks to find a liar's vertex-edge dominating set of $G$ of minimum cardinality. In this paper, we study this problem from an algorithmic point of view. We design two linear time algorithms for extsc{MinLVEDP} in block graphs and proper interval graphs, respectively. On the negative side, we show that the decision version of liar's vertex-edge domination problem is NP-complete for undirected path graphs.

Problem

• Studies liar's vertex-edge domination in chordal graphs.
• Develops linear algorithms for MinLVEDP in block and interval graphs.
• Proves NP-completeness of liar's vertex-edge domination in path graphs.

Innovation

• Linear time algorithm for block graphs
• Linear time algorithm for interval graphs
• NP-completeness proof for path graphs