Local Metric Dimension in Graphs

• Local metric dimension is a distance-based invariant that identifies the minimum number of vertices needed to differentiate adjacent vertices via distinct distance profiles.
• It is closely related to clique structures, graph products, and ILP formulations, providing both combinatorial and computational insights.
• Recent studies offer sharp bounds and fractional extensions, enhancing applications in network discovery, robot navigation, and mathematical chemistry.

Local metric dimension is a distance-based graph invariant that asks for the smallest number of vertices needed to distinguish adjacent vertices by their distance profiles. For a connected graph $G$, a set $W\subseteq V(G)$ is local resolving if for every adjacent $u,v\in V(G)$ there exists $w\in W$ such that $d_G(u,w)\neq d_G(v,w)$; a minimum such set is a local metric basis, and its cardinality is the local metric dimension, denoted in the literature by $lmd(G)$, $\dim_l(G)$, $\dim_\ell(G)$, or $\operatorname{ldim}(G)$ (Salman et al., 2014). This invariant is weaker than the classical metric dimension, because only adjacent pairs must be separated, but it remains tightly linked to clique structure, graph products, decomposition methods, integer programming, and fractional relaxations (Rodríguez-Velázquez et al., 2013).

1. Definitions and foundational relations

The common starting point is the distance function $d_G$ on a finite, simple, connected graph $W\subseteq V(G)$0. A vertex $W\subseteq V(G)$1 distinguishes $W\subseteq V(G)$2 and $W\subseteq V(G)$3 when

$W\subseteq V(G)$4

A local resolving set requires this only for adjacent pairs, whereas a resolving set for the ordinary metric dimension must distinguish all distinct pairs (Salman et al., 2014).

A closely related parameter is the local adjacency metric dimension. A set $W\subseteq V(G)$5 is a local adjacency resolving set if for every pair of adjacent vertices in $W\subseteq V(G)$6 there exists a vertex in $W\subseteq V(G)$7 adjacent to exactly one of them; its minimum cardinality is $W\subseteq V(G)$8 (Ghalavand et al., 18 Jul 2025). The comparison

$W\subseteq V(G)$9

is fundamental, because any upper bound on the adjacency-based variant transfers immediately to the local metric dimension (Ghalavand et al., 18 Jul 2025).

The principal parameters and their standard relations may be organized as follows.

Parameter	Meaning
$u,v\in V(G)$0	metric dimension
$u,v\in V(G)$1	adjacency dimension
$u,v\in V(G)$2	local metric dimension
$u,v\in V(G)$3	local adjacency dimension

The literature records the inequalities

$u,v\in V(G)$4

(Rodríguez-Velázquez et al., 2013). For a non-trivial connected graph of order $u,v\in V(G)$5, one also has

$u,v\in V(G)$6

where $u,v\in V(G)$7 denotes the metric dimension (Salman et al., 2014).

This hierarchy clarifies a common source of confusion: local metric dimension is not a merely notational variant of metric dimension. It is strictly weaker, because it addresses only adjacency-level distinguishability. It is also distinct from strong metric dimension, where a vertex $u,v\in V(G)$8 strongly resolves $u,v\in V(G)$9 if

$w\in W$0

and from fault-tolerant metric dimension, where a resolving set must remain resolving after removal of any one of its elements (Salman et al., 2014).

2. Extremal values and clique-sensitive structure

Several extremal characterizations recur across the literature. For a connected graph $w\in W$1, the local metric dimension satisfies

$w\in W$2

$w\in W$3

and

$w\in W$4

where $w\in W$5 is the clique number (Ghalavand et al., 18 Jul 2025). These facts already show that clique structure is not peripheral: it controls both maximality and near-maximality.

A lower bound recalled in recent clique-number papers is

$w\in W$6

again tying local metric dimension to $w\in W$7 (Ghalavand et al., 18 Jul 2025).

Exact computations on explicit graph families illustrate how the parameter departs from the full metric dimension while remaining structurally sharp. For the convex polytopes $w\in W$8 and $w\in W$9,

$d_G(u,w)\neq d_G(v,w)$0

whereas the same paper proves substantially larger strong metric dimensions: $d_G(u,w)\neq d_G(v,w)$1 These computations make explicit that local metric dimension can remain small on non-bipartite families for which stronger resolving notions grow linearly in $d_G(u,w)\neq d_G(v,w)$2 (Salman et al., 2014).

The clique number also interacts with older upper bounds. Before the sharp 2025 results, the literature already included the triangle-free estimate

$d_G(u,w)\neq d_G(v,w)$3

for $d_G(u,w)\neq d_G(v,w)$4, and the former general bound

$d_G(u,w)\neq d_G(v,w)$5

which was weaker than the later conjectured coefficient $d_G(u,w)\neq d_G(v,w)$6 (Ghalavand et al., 18 Jul 2025).

3. Algorithmic formulations and computational complexity

Local metric dimension admits an exact integer linear programming formulation. For $d_G(u,w)\neq d_G(v,w)$7, with $d_G(u,w)\neq d_G(v,w)$8 and $d_G(u,w)\neq d_G(v,w)$9, define

$lmd(G)$0

for adjacent pairs $lmd(G)$1, and binary variables

$lmd(G)$2

The model is

$lmd(G)$3

subject to

$lmd(G)$4

for every adjacent pair, together with

$lmd(G)$5

The formulation has $lmd(G)$6 binary variables and $lmd(G)$7 linear constraints, and it is exact: $lmd(G)$8 is a local resolving set if and only if the constraints are satisfied (Salman et al., 2014).

From a complexity standpoint, exact solvability is limited. The local adjacency dimension problem is NP-complete, even on planar instances, and assuming ETH there is no

$lmd(G)$9

algorithm solving it (Rodríguez-Velázquez et al., 2013). A decisive corona-product reduction then yields the same negative results for the local metric dimension: LocDim is NP-complete, and assuming ETH there is no

$\dim_l(G)$0

algorithm solving LocDim on graphs of order $\dim_l(G)$1 and size $\dim_l(G)$2 (Rodríguez-Velázquez et al., 2013).

The reduction is driven by the identity

$\dim_l(G)$3

valid for a connected graph $\dim_l(G)$4 of order $\dim_l(G)$5 and a non-trivial graph $\dim_l(G)$6. In particular,

$\dim_l(G)$7

so any sufficiently strong algorithm for local metric dimension would solve local adjacency dimension as well (Rodríguez-Velázquez et al., 2013).

These algorithmic results explain why much of the subject developed through exact formulas for special families, structural decompositions, and sharp upper bounds rather than through general-purpose polynomial-time algorithms.

4. Graph products and decomposition frameworks

A large portion of the theory is organized around graph constructions for which local metric dimension can be transferred, bounded, or decomposed.

Construction	Representative result
Corona product	$\dim_l(G)$8 (Rodríguez-Velázquez et al., 2013)
Generalized hierarchical product	$\dim_l(G)$9 under a basis-containment hypothesis (Klavžar et al., 2019)
Strong product	$\dim_\ell(G)$0 (Barragan-Ramirez et al., 2015)
Lexicographic product	$\dim_\ell(G)$1 is expressed via $\dim_\ell(G)$2, true twin classes, and $\dim_\ell(G)$3 (Barragán-Ramírez et al., 2016)

For corona products, two complementary descriptions coexist. One expresses the parameter directly through local adjacency dimension: $\dim_\ell(G)$4 (Rodríguez-Velázquez et al., 2013). Another reduces the problem to $\dim_\ell(G)$5: if the vertex of $\dim_\ell(G)$6 does not belong to any local metric basis of $\dim_\ell(G)$7, then

$\dim_\ell(G)$8

while if it does belong to a local metric basis of $\dim_\ell(G)$9, then

$\operatorname{ldim}(G)$0

for connected $\operatorname{ldim}(G)$1 of order $\operatorname{ldim}(G)$2 (Rodriguez-Velazquez et al., 2013). A common misconception is that coronas always scale with $\operatorname{ldim}(G)$3; in fact, the exact multiplier is governed either by $\operatorname{ldim}(G)$4 or by $\operatorname{ldim}(G)$5, depending on the formulation (Rodríguez-Velázquez et al., 2013).

For generalized hierarchical products $\operatorname{ldim}(G)$6, one has a distance formula separating the $\operatorname{ldim}(G)$7- and $\operatorname{ldim}(G)$8-coordinates, and under the hypothesis that $\operatorname{ldim}(G)$9 contains a local metric basis contained in $d_G$0,

$d_G$1

Special cases include the Cartesian product: $d_G$2 and, when $d_G$3 is bipartite,

$d_G$4

(Klavžar et al., 2019).

For strong products, the distance formula

$d_G$5

drives both universal and structure-sensitive estimates. Besides the general upper bound already listed, the theory includes exact infinite families. For any connected bipartite graph $d_G$6 and any integer $d_G$7,

$d_G$8

and for any connected bipartite graph $d_G$9 and any integer $W\subseteq V(G)$00,

$W\subseteq V(G)$01

(Barragan-Ramirez et al., 2015).

Decomposition methods go beyond products. For graphs built by point-attaching from primary subgraphs $W\subseteq V(G)$02, the general formula

$W\subseteq V(G)$03

reduces the global computation to contributions from the non-bipartite primary subgraphs (Rodríguez-Velázquez et al., 2014). In particular, if $W\subseteq V(G)$04 is the only non-bipartite primary subgraph, then

$W\subseteq V(G)$05

(Rodríguez-Velázquez et al., 2014). For subgraph-amalgamation under isometric embedding, the literature proves

$W\subseteq V(G)$06

with refinements via co-traversals and covers (Barragan-Ramirez et al., 2015).

5. Sharp upper bounds in 2025 and equality constructions

The central recent development is the confirmation of the clique-number conjecture asserting that for graphs with

$W\subseteq V(G)$07

the local metric dimension should satisfy

$W\subseteq V(G)$08

The decisive step was to prove the stronger adjacency version: $W\subseteq V(G)$09 for graphs with $W\subseteq V(G)$10 and $W\subseteq V(G)$11. Because

$W\subseteq V(G)$12

this immediately yields

$W\subseteq V(G)$13

in full generality (Ghalavand et al., 18 Jul 2025).

The proof is constructive and combinatorial. It partitions the vertex set into classes determined by maximal collections of induced subgraphs isomorphic to

$W\subseteq V(G)$14

where $W\subseteq V(G)$15 denotes the graph obtained from $W\subseteq V(G)$16 by removing $W\subseteq V(G)$17 edges incident with a common vertex. A local adjacency resolving set $W\subseteq V(G)$18 is then built through $W\subseteq V(G)$19 processes. At each stage, all but one or all but two vertices are selected from suitable clique-like induced subgraphs, and structural observations labeled (I)–(V) prevent the formation of larger cliques in the residual graph (Ghalavand et al., 18 Jul 2025). This suggests that the sharp coefficient $W\subseteq V(G)$20 is not merely extremal but reflects a persistent “almost-all-vertices from near-cliques” mechanism.

Equality is attained by infinitely many graphs. For integers $W\subseteq V(G)$21 and $W\subseteq V(G)$22, let $W\subseteq V(G)$23 be formed from $W\subseteq V(G)$24 disjoint copies of $W\subseteq V(G)$25 by identifying one vertex from each copy into a single common vertex. Then

$W\subseteq V(G)$26

and

$W\subseteq V(G)$27

while

$W\subseteq V(G)$28

Hence the upper bound is sharp (Ghalavand et al., 18 Jul 2025).

The small-clique cases were established separately and sharply. For $W\subseteq V(G)$29-free graphs, equivalently $W\subseteq V(G)$30, one has

$W\subseteq V(G)$31

which proves the conjectured $W\subseteq V(G)$32 case and resolves positively the planar-graph problem in the $W\subseteq V(G)$33 regime (Ghalavand et al., 31 May 2025). Equality holds for infinitely many planar graphs of odd order $W\subseteq V(G)$34, namely

$W\subseteq V(G)$35

for which

$W\subseteq V(G)$36

(Ghalavand et al., 31 May 2025).

For $W\subseteq V(G)$37-free graphs, the sharp bounds are

$W\subseteq V(G)$38

$W\subseteq V(G)$39

$W\subseteq V(G)$40

The $W\subseteq V(G)$41 case is obtained through a five-stage constructive argument based on maximum vertex-disjoint induced copies of six small graphs $W\subseteq V(G)$42, followed by isolated vertices $W\subseteq V(G)$43 (Ghalavand et al., 18 Jul 2025). The bounds are sharp for planar graphs; for example,

$W\subseteq V(G)$44

satisfies $W\subseteq V(G)$45, is planar, and attains

$W\subseteq V(G)$46

(Ghalavand et al., 18 Jul 2025). The 2025 global theorem subsumes these earlier cases while preserving the equality phenomenon (Ghalavand et al., 18 Jul 2025).

6. Fractional extensions, applications, and adjacent directions

A fractional relaxation replaces characteristic vectors of local resolving sets by weights. For an edge $W\subseteq V(G)$47, the local resolving neighborhood is

$W\subseteq V(G)$48

and a function $W\subseteq V(G)$49 is a local resolving function if

$W\subseteq V(G)$50

The minimum value of $W\subseteq V(G)$51 is the fractional local metric dimension $W\subseteq V(G)$52 (Benish et al., 2018). Equivalent notation using $W\subseteq V(G)$53 for adjacent pairs is also standard (Ali et al., 2021).

The fractional parameter satisfies

$W\subseteq V(G)$54

$W\subseteq V(G)$55

where

$W\subseteq V(G)$56

and

$W\subseteq V(G)$57

(Ali et al., 2021). A sharp extremal characterization is

$W\subseteq V(G)$58

and for vertex-transitive graphs,

$W\subseteq V(G)$59

(Benish et al., 2018).

Fractional local metric dimension has its own product theory. For strong products,

$W\subseteq V(G)$60

while for Cartesian products,

$W\subseteq V(G)$61

(Benish et al., 2018). On rotationally symmetric planar graphs obtained by edge coalescence of chorded cycles, exact values and asymptotically bounded families have been established; for instance,

$W\subseteq V(G)$62

and many pentagonal and hexagonal families have upper bounds tending to $W\subseteq V(G)$63 as $W\subseteq V(G)$64 (Ali et al., 2021). On Toeplitz and zero-divisor graphs, upper-bound sequences fall into constant, bounded, and unbounded families; examples of constant families include

$W\subseteq V(G)$65

and

$W\subseteq V(G)$66

(Alali et al., 2023).

Applications repeatedly cited for local metric dimension and its variants include network discovery and verification, robot navigation, chemistry, graphs arising in mathematical chemistry, and a delivery-services customer-coding model in which customers are assigned codes built from family-name initials and local metric representations (Salman et al., 2014). In that model, the advantage is that local metric dimension always satisfies

$W\subseteq V(G)$67

so local codes can be at least as compact as classical metric-dimension codes, and for bipartite graphs one has $W\subseteq V(G)$68 while $W\subseteq V(G)$69 can be arbitrarily large (Klavžar et al., 2019).

Taken together, these developments position local metric dimension as a structurally rich relaxation of metric dimension: it is weak enough to admit sharp clique-dependent bounds, product formulas, and compact ILP models, yet strong enough to encode nontrivial geometry, twin structure, and local identifiability across a wide range of graph classes (Ghalavand et al., 18 Jul 2025).