scholarly journals Computing the Domination Number of Grid Graphs

10.37236/628 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Samu Alanko ◽  
Simon Crevals ◽  
Anton Isopoussu ◽  
Patric Östergård ◽  
Ville Pettersson
Keyword(s):  
Dynamic Programming ◽  
Dominating Set ◽  
Domination Number ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Dominating Sets ◽  
Grid Graph ◽  
Grid Graphs ◽  
Square Grid ◽  
Minimum Dominating Set

Let $\gamma_{m,n}$ denote the size of a minimum dominating set in the $m \times n$ grid graph. For the square grid graph, exact values for $\gamma_{n,n}$ have earlier been published for $n \leq 19$. By using a dynamic programming algorithm, the values of $\gamma_{m,n}$ for $m,n \leq 29$ are here obtained. Minimum dominating sets for square grid graphs up to size $29 \times 29$ are depicted.

scholarly journals Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Mrinal Nandi ◽  
Subrata Parui ◽  
Avishek Adhikari
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Domination Number ◽  
Cartesian Product ◽  
Wireless Sensor ◽  
Dominating Sets ◽  
Grid Graph ◽  
Grid Graphs ◽  
Cylindrical Grid ◽  
Domination Numbers

Let γPm □ Cn denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs Pm, the path of length m, m≥2, and the graph Cn, the cycle of length n, n≥3. In this paper we propose methods to find the domination numbers of graphs of the form Pm □ Cn with n≥3 and m=5 and propose tight bounds on domination numbers of the graphs P6 □ Cn, n≥3. Moreover, we provide rough bounds on domination numbers of the graphs Pm □ Cn, n≥3 and m≥7. We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks.

On computing secure domination of trees

2020 ◽  
pp. 2150055
Author(s):  
Abolfazl Poureidi
Keyword(s):  
Linear Time ◽  
Search Algorithm ◽  
Dominating Set ◽  
Domination Number ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Time And Space ◽  
Minimum Cardinality ◽  
Secure Domination ◽  
Appl Math

Let [Formula: see text] be a graph. A subset [Formula: see text] is a dominating set of [Formula: see text] if for each [Formula: see text] there is a vertex [Formula: see text] adjacent to [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is a secure dominating set of [Formula: see text] if for each [Formula: see text] there is a vertex [Formula: see text] adjacent to [Formula: see text] such that [Formula: see text] is also a dominating set of [Formula: see text]. The minimum cardinality of a secure dominating set of [Formula: see text] is called the secure domination number of [Formula: see text]. Burger et al. [A linear algorithm for secure domination in trees, Discrete Appl. Math. 171 (2014) 15–27] proposed a nontrivial algorithm for computing a minimum secure dominating set of a given tree in linear time and space. In this paper, we give a dynamic programming algorithm to compute the secure domination number of a given tree [Formula: see text] in [Formula: see text] time and space and then using a backtracking search algorithm we can find a minimum secure dominating set of [Formula: see text] in [Formula: see text] time and space that its implementation is much simpler than the implementation of the algorithm proposed by Burger et al.

Optimization in Trajectory Planning of Multi-Jointed Fingers in Dextrous Hand Designs

1991 ◽  
Author(s):  
A. Meghdari ◽  
H. Sayyaadi
Keyword(s):  
Dynamic Programming ◽  
Motion Control ◽  
Trajectory Planning ◽  
Degrees Of Freedom ◽  
Optimization Technique ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Kinematics And Dynamics ◽  
Feasible Solutions ◽  
Dextrous Hand

Abstract An optimization technique based on the well known Dynamic Programming Algorithm is applied to the motion control trajectories and path planning of multi-jointed fingers in dextrous hand designs. A three fingered hand with each finger containing four degrees of freedom is considered for analysis. After generating the kinematics and dynamics equations of such a hand, optimum values of the joints torques and velocities are computed such that the finger-tips of the hand are moved through their prescribed trajectories with the least time or/and energy to reach the object being grasped. Finally, optimal as well as feasible solutions for the multi-jointed fingers are identified and the results are presented.

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