Gabriel单位圆盘图的度量维数是NP-完全的

单位圆盘图的最大割和最大二等分是NP-难的

Max-Cut and Max-Bisection are NP-hard on unit disk graphs

证明了单位圆盘图的最大割和最大二分问题是NP-难的。我们还证明了当$λ$ > 1 / \sqrt{2} $时，$λ$ -精度图是平面图.

Total Domination in Unit Disk Graphs

Let $G=(V,E)$ be an undirected graph. We call $D_t \subseteq V$ as a total dominating set (TDS) of $G$ if each vertex $v \in V$ has a dominator in $D$ other than itself. Here we consider the TDS problem in unit disk graphs, where the objective is to find a minimum cardinality total dominating set for an input graph. We prove that the TDS problem is NP-hard in unit disk graphs. Next, we propose an 8-factor approximation algorithm for the problem. The running time of the proposed approximation algorithm is $O(n \log k)$, where $n$ is the number of vertices of the input graph and $k$ is output si…

Liar's Domination in Unit Disk Graphs

In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. Next, we show that the recent sub-quadratic time $\frac{11}{2}$-factor approximation algorithm \cite{bhore} for the MLDS problem is erroneous and propose a simple $O(n + m)$ time 7.31-factor approximation algorithm, where $n$ and $m$ are the number of vertices and edges in the input unit disk graph, respectively. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.

带圆盘的刺线段：复杂性和近似算法

Stabbing line segments with disks: complexity and approximation algorithms

计算的复杂性和近似算法的报告问题的刺一组直线段的最小基数集的固定半径的磁盘$r>0$，其中的一组段形成一条直线画$G=(V,E)$的平面图没有边交叉。在网络安全应用中，经常出现闭几何问题.本文给出了Delaunay三角剖分、Gabriel图和其它子图（常用于网络设计）的边集问题对于$r\in [d_{\min},ηd_{\max}]$和某些常数$η$的强NP-困难性，其中$d_{\max}$和$d_{\min}$分别是图的最长边和最短边的欧氏长度.本文在平面图的直线图类中提出了一种快速$O(|E|\log|E|)$ -时间$O(1)$ -近似算法，对于这类平面图，不等式$r\geq ηd_{\max}$对某个常数$η>0,$一致成立，即当$G$的边长由$η>0,$的某个线性函数从上一致有界时

The Generalized Independent and Dominating Set Problems on Unit Disk Graphs

In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.

Firefighter Problem with Minimum Budget: Hardness and Approximation Algorithm for Unit Disk Graphs

Unit disk graphs are the set of graphs which represent the intersection of disk graphs and interval graphs. These graphs are of great importance due to their structural similarity with wireless communication networks. Firefighter problem on unit disk graph is interesting as it models the virus spreading in an wireless network and asks for a solution to stop it. In this paper, we consider the MIN-BUDGET firefighter problem where the goal is to determine the minimum number of firefighters required and the nodes to place them at each time instant to save a given set of vertices of a given graph a…