We consider, in this paper, the NP-hard problem of finding the minimum connected domination metric dimension of graphs. A vertex set B of a connected graph G = (V, E) resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. A resolving set B of G is connected if the subgraph induced by B is a nontrivial connected subgraph of G. A resolving set is dominating if every vertex of G that does not belong to B is a neighbor to some vertices in B. The cardinality of the smallest resolving set of G, the cardinality of the minimal connected resolving set, and the cardinality of the minimal connected domination resolving set are the metric dimension of G, connected metric dimension of G, and connected domination metric dimension of G, respectively. We present the first attempt to compute heuristically the minimum connected dominant resolving set of graphs by a binary version of the equilibrium optimization algorithm (BEOA). The particles of BEOA are binary-encoded and used to represent which one of the vertices of the graph belongs to the connected domination resolving set. The feasibility is enforced by repairing particles such that an additional vertex generated from vertices of G is added to B, and this repairing process is iterated until B becomes the connected domination resolving set. The proposed BEOA is tested using graph results that are computed theoretically and compared to competitive algorithms. Computational results and their analysis show that BEOA outperforms the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWO), the binary Slime Mould Optimizer (BSMO), the binary Grasshopper Optimizer (BGO), the binary Artificial Ecosystem Optimizer (BAEO), and the binary Elephant Herding Optimizer (BEHO).

In this paper, we consider the NP-hard problem of finding the minimum connected resolving set of graphs. A vertex set B of a connected graph G resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. A resolving set B of G is connected if the subgraph induced by B is a nontrivial connected subgraph of G. The cardinality of the minimal resolving set is the metric dimension of G and the cardinality of minimum connected resolving set is the connected metric dimension of G. The problem is solved heuristically by a binary version of an enhanced Harris Hawk Optimization (BEHHO) algorithm. This is the first attempt to determine the connected resolving set heuristically. BEHHO combines classical HHO with opposition-based learning, chaotic local search and is equipped with an S-shaped transfer function to convert the continuous variable into a binary one. The hawks of BEHHO are binary encoded and are used to represent which one of the vertices of a graph belongs to the connected resolving set. The feasibility is enforced by repairing hawks such that an additional node selected from V\B is added to B up to obtain the connected resolving set. The proposed BEHHO algorithm is compared to binary Harris Hawk Optimization (BHHO), binary opposition-based learning Harris Hawk Optimization (BOHHO), binary chaotic local search Harris Hawk Optimization (BCHHO) algorithms. Computational results confirm the superiority of the BEHHO for determining connected metric dimension.

In this paper, we consider the NP-hard problem of finding the minimum dominant resolving set of graphs. A vertex set B of a connected graph G resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. A resolving set is dominating if every vertex of G that does not belong to B is a neighbor to some vertices in B. The dominant metric dimension of G is the cardinality number of the minimum dominant resolving set. The dominant metric dimension is computed by a binary version of the Archimedes optimization algorithm (BAOA). The objects of BAOA are binary encoded and used to represent which one of the vertices of the graph belongs to the dominant resolving set. The feasibility is enforced by repairing objects such that an additional vertex generated from vertices of G is added to B and this repairing process is iterated until B becomes the dominant resolving set. This is the first attempt to determine the dominant metric dimension problem heuristically. The proposed BAOA is compared to binary whale optimization (BWOA) and binary particle optimization (BPSO) algorithms. Computational results confirm the superiority of the BAOA for computing the dominant metric dimension.

The law can be considered an important tool to address the risks of using artificial intelligence (AI). AI is defined in a variety of ways depending on the tasks it completes. Given that AI leverages computing power to carry out tasks that people typically undertake, it is also frequently referred to as cognitive computing or machine learning. Artificial intelligence (AI) uses data perception and synthesis to replicate human thought processes, automate tasks, and make judgments. The use of AI is regulated by many laws and regulations aimed at protecting consumers, users and society in general. The role of the law in addressing the risks of using AI includes many issues, among which are: maintaining privacy and security, maintaining fairness, civil and criminal liability, maintain safety and regulating the use of AI in business. Artificial intelligence in law firms has proven to be a golden ticket to increased productivity, improved decision-making, and higher competitiveness in the industry. Rules that individuals and organizations must adhere to when using AI, ensuring that these standards are strictly applied. Furthermore, the law helps promote transparency and accountability, as organizations have to commit to documenting AI usage processes and clarify how data and algorithms are used. This helps reduce the risk of discrimination and errors that can occur when using the AI.

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.