# Dominant Mixed Metric Dimension of Graph

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.### References

J. L. Gross, J. Yellen and P. Zhang. {\it Handbook of graph Theory} Second Edition CRC Press Taylor and Francis Group, (2014).

G. Chartrand and L. Lesniak. {\it Graphs and digraphs} {\it 3rd} ed (London: Chapman and Hall), (2000).

N. Hartsfield dan G. Ringel. \textit{Pearls in Graph Theory} Academic Press. United Kingdom, (1994).

P.J. Slater, Leaves of trees, in:\textit{Proc. 6th Southeast Conf. Comb., Graph Theory, Comput. Boca Rotan}, {\bf 14} (1975), 549-559.

F. Harary and R.A. Melter, On The metric dimension of a graph, \textit{Ars Combin}, {\bf 2} (1976), 191-195.

Haynes, T.W., Hedetniemi, S. and Slater, P., 1998. Fundamentals of domination in graphs. CRC Press.

Chartrand, G., Eroh, L., Johnson, M. A., & Oellermann, O. R. (2000). Resolvability in graphs and the metric dimension of a graph. Discrete Applied Mathematics, 105(1-3), 99-113.

Kelenc, A., Tratnik, N., and Yero, I. G. (2018). Uniquely identifying the edges of a graph: the edge metric dimension. Discrete Applied Mathematics, 251, 204-220.

Kelenc, A., Kuziak, D., Taranenko, A., & Yero, I. G. (2017). Mixed metric dimension of graphs. Applied Mathematics and Computation, 314, 429-438.

G. Chartrand and L. Lesniak. {\it Graphs and digraphs} {\it 3rd} ed (London: Chapman and Hall), (2000).

N. Hartsfield dan G. Ringel. \textit{Pearls in Graph Theory} Academic Press. United Kingdom, (1994).

P.J. Slater, Leaves of trees, in:\textit{Proc. 6th Southeast Conf. Comb., Graph Theory, Comput. Boca Rotan}, {\bf 14} (1975), 549-559.

F. Harary and R.A. Melter, On The metric dimension of a graph, \textit{Ars Combin}, {\bf 2} (1976), 191-195.

Haynes, T.W., Hedetniemi, S. and Slater, P., 1998. Fundamentals of domination in graphs. CRC Press.

Chartrand, G., Eroh, L., Johnson, M. A., & Oellermann, O. R. (2000). Resolvability in graphs and the metric dimension of a graph. Discrete Applied Mathematics, 105(1-3), 99-113.

Kelenc, A., Tratnik, N., and Yero, I. G. (2018). Uniquely identifying the edges of a graph: the edge metric dimension. Discrete Applied Mathematics, 251, 204-220.

Kelenc, A., Kuziak, D., Taranenko, A., & Yero, I. G. (2017). Mixed metric dimension of graphs. Applied Mathematics and Computation, 314, 429-438.

Published

2024-06-21

How to Cite

*Statistics, Optimization & Information Computing*. https://doi.org/10.19139/soic-2310-5070-1925

Issue

Section

Research Articles

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).