Research Article
Some New Results on Domination and Independent Dominating Set of Some Graphs
Basma Mohamed*
,
Mohammed Badawy
Issue:
Volume 13, Issue 3, June 2024
Pages:
53-57
Received:
22 February 2024
Accepted:
12 March 2024
Published:
10 May 2024
Abstract: One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set S⊆V is said to be dominating set of G if for every v є V-S there exists a vertex u є S such that uv є E. The dominance number of G, represented by γ(G), is the lowest cardinality of vertices among the dominating set of G. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph G and input K, γ(G) ≤ K. This is known as the dominating set issue. Consequently, it is thought that calculating γ(G) for each given graph G may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset S, then S⊆V is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of G is a set S of vertices in a graph G that is both an independent and a dominating set of G. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others.
Abstract: One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set S⊆V is said...
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Research Article
Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method
Muhammad Toriqul Islam,
Muhammad Shakhawat Hossain*
Issue:
Volume 13, Issue 3, June 2024
Pages:
58-68
Received:
2 February 2024
Accepted:
4 March 2024
Published:
3 June 2024
Abstract: This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, the triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is O(hp+2) superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.
Abstract: This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization...
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