engleski

Extremal mixed metric dimension with respect to the cyclomatic number

In a graph the cardinality of the smallest ordered set of vertices that distinguishes every element of is called the mixed metric dimension of and it is denoted by In [12] it was conjectured that every graph with cyclomatic number satisfies where is the number of leaves in . It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree . In this paper we determine that for every -graph the mixed metric dimension equals 3 or 4, with 4 being attained if and only if is a balanced -graph. Thus, for balanced -graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality holds.

mixed metric dimension ; unicyclic graphs ; cactus graphs

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