Local face metric dimension: a new resolvability parameter for Planar graphs

Amal S. Alali; Furqan Ahmad; Zubair Hafeez; Refah Alotaibi; Amal S. Alali; Furqan Ahmad; Zubair Hafeez; Refah Alotaibi

doi:10.3934/math.2026585

AIMS Mathematics

2026, Volume 11, Issue 5: 14253-14269. doi: 10.3934/math.2026585

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Local face metric dimension: a new resolvability parameter for Planar graphs

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Riphah International University, Lahore, Pakistan
3.
Department of Computer Science, Rowan University, United States

Received: 14 January 2026 Revised: 17 April 2026 Accepted: 06 May 2026 Published: 19 May 2026
MSC : 05C10, 05C12

Resolving sets and metric-based parameters capture the ability to distinguish elements of a graph through distances. While the classical metric dimension focuses on distinguishing vertices, planar graphs naturally suggest an analogous question at the level of faces. In this work we introduce the local face metric dimension, a new parameter that measures the least number of vertices needed so that every two adjacent faces have distinct distance representation. This concept extends the scope of resolvability theory from vertex-sets to the facial structure of planar graphs. We develop the basic theory of local face metric dimension for connected planar graphs, focusing on its structural properties and the conditions under which it is well defined and finite. The parameter reflects a localized notion of resolvability and highlights the interaction between graph distances and the facial structure of planar embeddings. Its behavior is explored for several standard graph families to illustrate its variability and dependence on graph structure. The local face metric dimension thus opens a new direction in the study of graph resolvability, highlighting the interplay between metric properties and planar embeddings, and suggesting further structural, extremal, and algorithmic investigations.
- face resolving set,
- face metric dimension,
- local face resolving set,
- local face metric dimension
Citation: Amal S. Alali, Furqan Ahmad, Zubair Hafeez, Refah Alotaibi. Local face metric dimension: a new resolvability parameter for Planar graphs[J]. AIMS Mathematics, 2026, 11(5): 14253-14269. doi: 10.3934/math.2026585

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Abstract

Resolving sets and metric-based parameters capture the ability to distinguish elements of a graph through distances. While the classical metric dimension focuses on distinguishing vertices, planar graphs naturally suggest an analogous question at the level of faces. In this work we introduce the local face metric dimension, a new parameter that measures the least number of vertices needed so that every two adjacent faces have distinct distance representation. This concept extends the scope of resolvability theory from vertex-sets to the facial structure of planar graphs. We develop the basic theory of local face metric dimension for connected planar graphs, focusing on its structural properties and the conditions under which it is well defined and finite. The parameter reflects a localized notion of resolvability and highlights the interaction between graph distances and the facial structure of planar embeddings. Its behavior is explored for several standard graph families to illustrate its variability and dependence on graph structure. The local face metric dimension thus opens a new direction in the study of graph resolvability, highlighting the interplay between metric properties and planar embeddings, and suggesting further structural, extremal, and algorithmic investigations.

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