# Dominant Mixed Metric Dimension of Graph

• Ridho Alfarisi UNIVERSITAS JEMBER
• Sharifah Kartini Said Husain Institute of Mathematical Research, Universiti Putra Malaysia, Malaysia
• Liliek Susilowati Department of Mathematics, Universiti Airlangga, Indonesia
• Arika Indah Kristiana Department of Mathematics Education, University of Jember, Indonesia
Keywords: Mixed resolving set; dominating set; mixed resolving dominating set; dominant mixed metric dimension

### Abstract

For $k-$ordered set $W=\{s_1, s_2,\dots, s_k \}$ of vertex set $G$, the representation of a vertex or edge $a$ of $G$ with respect to $W$ is $r(a|W)=(d(a,s_1), d(a,s_2),\dots, d(a,s_k))$ where $a$ is vertex so that $d(a,s_i)$ is a distance between of the vertex $v$ and the vertices in $W$ and $a=uv$ is edge so that $d(a,s_i)=min\{d(u,s_i),d(v,s_i)\}$. The set $W$ is a mixed resolving set of $G$ if $r(a|W)\neq r(b|W)$ 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 mixed metric dimension, denoted by $dim_m(G)$. 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 dominating set is domination number , denoted by $\gamma(G)$. 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 mixed resolving dominating set is called dominant mixed metric dimension, denoted by $\gamma_{mr}(G)$. In our paper, we will investigated the establish sharp bounds of the dominant mixed metric dimension of $G$ and determine the exact value of some family graphs.

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Published
2024-06-21
How to Cite
Alfarisi, R., Husain, S. K. S., Susilowati, L., & Kristiana, A. I. (2024). Dominant Mixed Metric Dimension of Graph. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-1925
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Section
Research Articles