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The distance d(va,vb) between two vertices of a simple connected graph G is the length of the shortest path between va and vb. Vertices va,vb of G are considered to be resolved by a vertex v if d(va,v)≠d(vb,v). An ordered set W={v1,v2,v3,…,vs}⊆V(G) is said to be a resolving set for G, if for any va,vb∈V(G),∃vi∈W∋d(va,vi)≠d(vb,vi). The representatio...
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... k,l,m obtained from three pairwise internal disjoint paths P k , P l , and P m , by joining starting vertices of P k and P m to the starting vertex of P l , and ending vertices of P k and P m , to the ending vertex of P l . Let us denote the vertices of this graph as v 1 , v 2 , · · · , v k+l+m , then this type of bicyclic graph is given in Figure 3. Note that the starting vertices of paths, i.e., v 1 of P k , v k+1 of P l , and v k+l+1 of P m , are joined together. ...
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... For a number of graphs in the literature, the metric dimension, domination metric dimension, and independent domination metric dimension are all theoretically specified. Following is a brief summary of the key recently discovered theoretical metric dimension results [17][18][19][20][21][22][23][24][25]. The metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree is determined theoretically in [17], the bary centric subdivision of Möbius ladders and the generalized Petersen multigraphs in [18], trapezoid network, − ( ) network, open ladder network, tortoise network in [19], French windmill graph and Dutch windmill graph in [20], total graph of path power three and four in [21], two types of bicyclic graphs in [22], Mobius Ladder in [23], power of paths and complement of paths in [24], and Kayak Paddles graph and Cycles with chord in [25]. ...
... Following is a brief summary of the key recently discovered theoretical metric dimension results [17][18][19][20][21][22][23][24][25]. The metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree is determined theoretically in [17], the bary centric subdivision of Möbius ladders and the generalized Petersen multigraphs in [18], trapezoid network, − ( ) network, open ladder network, tortoise network in [19], French windmill graph and Dutch windmill graph in [20], total graph of path power three and four in [21], two types of bicyclic graphs in [22], Mobius Ladder in [23], power of paths and complement of paths in [24], and Kayak Paddles graph and Cycles with chord in [25]. ...
In this article, we look at the NP-hard problem of determining the minimum independent domination metric dimension of graphs. A vertex set of a connected graph, resolves if every vertex of is uniquely recognized by its vector of distances to the vertices in. If there are no neighboring vertices in a resolving set of , then is independent. Every vertex of that does not belong to must be a neighbor of at least one vertex in for a resolving set to be dominant. The metric dimension of , independent metric dimension of , and independent dominant metric dimension of are, respectively, the cardinality of the smallest resolving set of , the minimal independent resolving set, and the minimal independent domination resolving set. We propose the first attempt to use a binary version of the Rat Swarm Optimizer Algorithm (BRSOA) to heuristically calculate the smallest independent dominant resolving set of graphs. The search agent of BRSOA are binary-encoded and used to identify which one of the vertices of the graph belongs to the independent domination resolving set. The feasibility is enforced by repairing search agent such that an additional vertex created from vertices of is added to , and this repairing process is repeated until becomes the independent domination resolving set. Using theoretically computed graph findings and comparisons to competing methods, the proposed BRSOA is put to the test. BRSOA surpasses the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWOA), the binary Gravitational Search Algorithm (BGSA), and the binary Moth-Flame Optimization (BMFO), according to computational results and their analysis.
