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- W3117317657 abstract "• In the second revision of the paper we have corrected the paper according to the suggestions of the reviewer. This includes slight revision of two proofs according to remarks 2. and 6. of the reviewer, while other remarks are mainly mistypes which are accordingly corrected. We thank the reviewer for the careful reading of the paper. In a graph G , cardinality of the smallest ordered set of vertices that distinguishes every element of V ( G ) is the (vertex) metric dimension of G . Similarly, the cardinality of such a set is the edge metric dimension of G , if it distinguishes E ( G ) . In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ by at most one for a same unicyclic graph. Next we extend the results to graphs with edge disjoint cycles (i.e. cactus graphs) showing that the two invariants can differ by at most c , where c is the number of cycles in such a graph. We conclude the paper with a conjecture that generalizes the previously mentioned consequences to graphs with prescribed cyclomatic number c by claiming that the difference of the invariant is still bounded by c ." @default.
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- W3117317657 date "2021-05-01" @default.
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- W3117317657 title "Bounds on metric dimensions of graphs with edge disjoint cycles" @default.
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- W3117317657 doi "https://doi.org/10.1016/j.amc.2020.125908" @default.
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