Abstract
A vertex w of a connected graph G strongly resolves two distinct vertices u,v∈V(G), if there is a shortest u,w path containing v, or a shortest v,w path containing u. A set S of vertices of G is a strong resolving set for G if every two distinct vertices of G are strongly resolved by a vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. To study the strong metric dimension of graphs, a very important role is played by a structure of graphs called the strong resolving graph In this work, we obtain the strong metric dimension of some families of cactus graphs, and along the way, we give several structural properties of the strong resolving graphs of the studied families of cactus graphs.
Highlights
Topics concerning metric dimension and related parameters in graphs are nowadays very common in the research community, probably based on its applicability to diverse practical problems of identification of nodes in networks
Despite the fact that the bounds above are proved, we might notice that the problem of describing the strong resolving graph, and of computing the strong metric dimension of cactus graphs seems to be very challenging based on the situation that we can not control things like the orders of the involved cycles, the number of terminal vertices and cut vertices, their adjacencies, etc
We are centered into computing or bounding the strong metric dimension of the cactus graphs which we have studied in the previous section
Summary
Topics concerning metric dimension and related parameters in graphs are nowadays very common in the research community, probably based on its applicability to diverse practical problems of identification of nodes in networks. A fairly complete study on results and open questions concerning the strong metric dimension of graphs can be found in [8]. To see this relationship, for a given graph G, the construction of a new related graph, called strong resolving graph, was required. Where the authors presented some general results for the strong metric dimension of corona product graph and rooted product graphs, respectively Clear definitions of these two graph products can be found in [8]. We begin to formalize all the required notations and terminologies that shall be used throughout the document To this end, for the whole exposition, let G be a connected simple graph with vertex set V ( G ). S ⊂ V ( G ), by hSi we represent the subgraph induced by S in G
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