Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTotal Domination in Unit Disk Graphs
72
0
0.0
(
0
)
Added by
Sangram Kishor Jena Mr
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 size. We also show that TDS problem admits a PTAS in unit disk graphs.
rate research
Read More
In this article, we study a variant of the minimum dominating set problem known as the minimum liars 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.
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.
Greedy routing has been studied successfully on Euclidean unit disk graphs, which we interpret as a special case of hyperbolic unit disk graphs. While sparse Euclidean unit disk graphs exhibit grid-like structure, we introduce strongly hyperbolic unit disk graphs as the natural counterpart containing graphs that have hierarchical network structures. We develop and analyze a routing scheme that utilizes these hierarchies. On arbitrary graphs this scheme guarantees a worst case stretch of $max{3, 1+2b/a}$ for $a > 0$ and $b > 1$. Moreover, it stores $mathcal{O}(k(log^2{n} + log{k}))$ bits at each vertex and takes $mathcal{O}(k)$ time for a routing decision, where $k = pi e (1 + a)/(2(b - 1)) (b^2 text{diam}(G) - 1) R + log_b(text{diam}(G)) + 1$, on strongly hyperbolic unit disk graphs with threshold radius $R > 0$. In particular, for hyperbolic random graphs, which have previously been used to model hierarchical networks like the internet, $k = mathcal{O}(log^2{n})$ holds asymptotically almost surely. Thus, we obtain a worst-case stretch of $3$, $mathcal{O}(log^4 n)$ bits of storage per vertex, and $mathcal{O}(log^2 n)$ time per routing decision on such networks. This beats existing worst-case lower bounds. Our proof of concept implementation indicates that the obtained results translate well to real-world networks.
For a graph $G=(V,E)$, we call a subset $ Ssubseteq V cup E$ a total mixed dominating set of $G$ if each element of $V cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $gamma_{tm}(G)$ of $G$ is the minimum cardinality of a total mixed dominating set of $G$. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number of a graph and its diameter. Studing of this number in trees is our next work. Also we show that the total mixed domination number of a graph is equale to the total domination number of a graph which is obtained by the graph. Giving the total mixed domination numbers of some special graphs is our last work.
A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks. Because the minimum dominating set problem for unit disk graphs is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTAS. However, since the proposal of a linear-time 5-approximation algorithm in 1995, the lack of efficient algorithms attaining better approximation factors has aroused attention. We introduce a linear-time O(n+m) approximation algorithm that takes the usual adjacency representation of the graph as input and outputs a 44/9-approximation. This approximation factor is also attained by a second algorithm, which takes the geometric representation of the graph as input and runs in O(n log n) time regardless of the number of edges. Additionally, we propose a 43/9-approximation which can be obtained in O(n^2 m) time given only the graphs adjacency representation. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
