Dominant mixed metric dimension of graph

For k−ordered set W = {s1, s2, …, sk} of vertex set G, the representation of a vertex or edge a of G with respect to W is r(a|W) = (d(a, s1), d(a, s2), …, d(a, sk)) where a is vertex so that d(a, si) is a distance between the vertex a and the vertices in W and a = uv is edge so that d(a, si) = min{d(u, si), d(v, si)}. The set W is a mixed resolving set of G if r(a|W) ≠ r(b|W) for every pair a, b of distinct vertices or edge of G. The minimum mixed resolving set W is a mixed basis of G. If G has a mixed basis, then its cardinality is called a mixed metric dimension, denoted by dimm(G). A set W of vertices in G is a dominating set for G if every vertex of G that is not in W is adjacent to some vertex of W. The minimum cardinality of the dominant set is the domination number, denoted by γ(G). A vertex set of some vertices in G that is both mixed resolving and dominating set is a mixed resolving dominating set. The minimum cardinality of the dominant set with mixed resolving is called the dominant mixed metric dimension, denoted by γmr(G). In our paper, we investigate the establishment of sharp bounds of the dominant mixed metric dimension of G and determine the exact value of some family graphs.

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