Local Adjacency Metric Dimension

• Local adjacency metric dimension is a graph invariant defined as the minimum size of a landmark set that distinguishes adjacent vertices solely by their neighborhood connections.
• It connects local resolving theory with adjacency-based metric theory, offering exact formulas for corona and lexicographic products as well as sharp clique number bounds.
• Recent studies highlight its NP-completeness and structural dichotomies, establishing its significance in both extremal graph theory and product-graph analysis.

Searching arXiv for recent and foundational papers on local adjacency metric dimension and related parameters. Local adjacency metric dimension is a graph invariant that measures the minimum size of a landmark set that distinguishes only adjacent vertices and does so by adjacency rather than by full shortest-path distance. In the literature, the same parameter appears as local adjacency dimension $\operatorname{adim}_\ell(G)$ and as local adjacency metric dimension $\dim_{A,l}(G)$. Formally, for a graph $G$, a set $S\subseteq V(G)$ is a local adjacency generator if for every two adjacent vertices $x,y\in V(G)\setminus S$, there exists $s\in S$ adjacent to exactly one of $x$ and $y$; a minimum such set is a local adjacency basis, and its cardinality is the invariant. The subject connects local resolving theory, adjacency-based metric theory, graph products, and structural bounds in terms of classical invariants such as clique number (Rodríguez-Velázquez et al., 2013, Barragán-Ramírez et al., 2016, Ghalavand et al., 18 Jul 2025).

1. Definitions and terminological variants

The parameter is defined on finite simple graphs. In the notation used in one line of work, a set $S\subseteq V(G)$ is a local adjacency generator if for every two adjacent vertices $x,y\in V(G)\setminus S$, there exists $\dim_{A,l}(G)$0 such that

$\dim_{A,l}(G)$1

A minimum local adjacency generator is a local adjacency basis, and its size is denoted

$\dim_{A,l}(G)$2

Another line of work uses the notation $\dim_{A,l}(G)$3 and the name local adjacency dimension for the same adjacency-based local resolving condition. In that terminology, the concepts of local adjacency generator, local adjacency basis, and local adjacency dimension are “defined by analogy” with adjacency dimension and local metric dimension (Rodríguez-Velázquez et al., 2013, Barragán-Ramírez et al., 2016).

This parameter sits naturally beside the truncated-distance viewpoint of adjacency dimension. For a graph $\dim_{A,l}(G)$4, the truncated distance

$\dim_{A,l}(G)$5

encodes whether vertices are equal, adjacent, or non-adjacent. Adjacency dimension is the metric dimension of $\dim_{A,l}(G)$6, and local adjacency dimension is the corresponding local analogue in which only adjacent pairs must be distinguished. A common misconception is that “local adjacency metric dimension” denotes a different invariant from “local adjacency dimension”; in the cited literature, the distinction is terminological rather than mathematical, and the local parameter is the adjacency-based counterpart of local metric dimension (Barragán-Ramírez et al., 2016, Estrada-Moreno et al., 2015).

2. Position within resolving theory

The local adjacency metric dimension is one member of a four-parameter family obtained by crossing two distinctions: all pairs versus adjacent pairs, and full distances versus adjacency-truncated information. The standard inequalities recorded in the literature are

$\dim_{A,l}(G)$7

$\dim_{A,l}(G)$8

$\dim_{A,l}(G)$9

$G$0

The inequality

$G$1

is especially important: if a vertex is adjacent to exactly one of two adjacent vertices, then its distances to them differ, so every local adjacency generator is automatically a local metric generator (Rodríguez-Velázquez et al., 2013, Ghalavand et al., 18 Jul 2025).

The diameter-two regime collapses the metric and adjacency local parameters. One source states that if $G$2, then

$G$3

and another states, in the same spirit, that if $G$4 has diameter two, then

$G$5

Thus the distinction between metric and adjacency local resolving is meaningful mainly beyond diameter two (Barragán-Ramírez et al., 2016, Rodríguez-Velázquez et al., 2013).

The parameter behaves differently from local metric dimension on basic graph classes. For paths $G$6 with $G$7,

$G$8

For stars $G$9 with $S\subseteq V(G)$0,

$S\subseteq V(G)$1

These formulas show that local adjacency dimension can coincide with local metric dimension on one family and separate sharply from it on another. The parameter is also bounded above by the vertex cover number: $S\subseteq V(G)$2 because every vertex cover is a local adjacency generator (Rodríguez-Velázquez et al., 2013).

3. Corona products and exact families

The foundational product formula for the subject concerns corona products. If $S\subseteq V(G)$3 is a connected graph of order $S\subseteq V(G)$4 and $S\subseteq V(G)$5 is a non-trivial graph, then

$S\subseteq V(G)$6

This identity turns a local metric problem on $S\subseteq V(G)$7 into a local adjacency problem on $S\subseteq V(G)$8, and it is the main bridge by which local adjacency dimension entered product-graph theory (Rodríguez-Velázquez et al., 2013).

The same source gives a dichotomy for the local adjacency dimension of the corona product itself. If there exists a local adjacency basis $S\subseteq V(G)$9 for $x,y\in V(G)\setminus S$0 such that for every $x,y\in V(G)\setminus S$1,

$x,y\in V(G)\setminus S$2

then

$x,y\in V(G)\setminus S$3

If, on the contrary, for any local adjacency basis $x,y\in V(G)\setminus S$4 for $x,y\in V(G)\setminus S$5, there exists some $x,y\in V(G)\setminus S$6 such that

$x,y\in V(G)\setminus S$7

then

$x,y\in V(G)\setminus S$8

Accordingly, for connected $x,y\in V(G)\setminus S$9 of order $s\in S$0 and non-trivial $s\in S$1, either

$s\in S$2

or

$s\in S$3

The same theory yields exact formulas on standard families. For $s\in S$4,

$s\in S$5

and

$s\in S$6

For a non-empty graph $s\in S$7 of order $s\in S$8,

$s\in S$9

if and only if $x$0 is a bipartite graph having only one non-trivial connected component $x$1 and $x$2, while

$x$3

if and only if

$x$4

Special corona formulas include

$x$5

and

$x$6

These results show that the local adjacency parameter is already rich on very elementary families (Rodríguez-Velázquez et al., 2013).

4. Lexicographic products and fiberwise structure

The lexicographic product provides the main setting in which local adjacency dimension acts as a fiberwise building block for other invariants. For a connected graph $x$7 of order $x$8, an ordered family $x$9, and non-singleton true twin classes $y$0 of $y$1, the local metric dimension of $y$2 is expressed by

$y$3

Here the first term is the baseline contribution from the fibers, the second corrects for true twin classes, and the third is a residual correction term governed by a relation $y$4 on certain vertices of the base graph. The central mechanism is the distance formula inside a fiber: $y$5 which forces local adjacency dimension, rather than local metric dimension of $y$6, to control same-fiber adjacent pairs (Barragán-Ramírez et al., 2016).

The same paper also derives the local adjacency dimension of the lexicographic product itself: $y$7 Thus the local metric and local adjacency dimensions of $y$8 differ only through the replacement of $y$9 by $S\subseteq V(G)$0. The criterion

$S\subseteq V(G)$1

makes this relation explicit. When all fibers are non-empty, the local metric formula simplifies to

$S\subseteq V(G)$2

and for complete base graphs,

$S\subseteq V(G)$3

These formulas show that local adjacency dimension is not merely an auxiliary notion: it is the primary fiber invariant governing lexicographic-product behavior (Barragán-Ramírez et al., 2016).

5. Clique number and sharp upper bounds

A recent development links local adjacency metric dimension directly to clique number. If $S\subseteq V(G)$4 is a simple connected graph with

$S\subseteq V(G)$5

then

$S\subseteq V(G)$6

Because

$S\subseteq V(G)$7

this immediately confirms the conjectured upper bound for local metric dimension as well. The theorem is sharp: for integers $S\subseteq V(G)$8 and $S\subseteq V(G)$9, if $x,y\in V(G)\setminus S$0 is obtained from $x,y\in V(G)\setminus S$1 copies of $x,y\in V(G)\setminus S$2 by identifying one chosen vertex from each copy into a common vertex, then

$x,y\in V(G)\setminus S$3

$x,y\in V(G)\setminus S$4

and

$x,y\in V(G)\setminus S$5

so equality holds in the bound (Ghalavand et al., 18 Jul 2025).

The proof is constructive and combinatorial. It partitions $x,y\in V(G)\setminus S$6 into maximal induced subgraphs of types $x,y\in V(G)\setminus S$7 and $x,y\in V(G)\setminus S$8, derives structural restrictions from maximality and the clique bound, and then constructs a local adjacency generator $x,y\in V(G)\setminus S$9 by selecting almost all vertices from each block while preserving adjacency distinguishability through $\dim_{A,l}(G)$00-configurations. The counting argument is tuned to the coefficient

$\dim_{A,l}(G)$01

For small clique number the bound specializes to familiar forms: $\dim_{A,l}(G)$02

$\dim_{A,l}(G)$03

The case $\dim_{A,l}(G)$04, equivalently the $\dim_{A,l}(G)$05-free setting, had already been isolated on the local metric side. If $\dim_{A,l}(G)$06 is a graph with $\dim_{A,l}(G)$07, then

$\dim_{A,l}(G)$08

which confirms the clique-number conjecture for $\dim_{A,l}(G)$09 and, in particular, yields the same half-order upper bound for planar graphs with $\dim_{A,l}(G)$10 (Ghalavand et al., 31 May 2025, Ghalavand et al., 18 Jul 2025).

6. Complexity and current directions

The decision problem LocAdjDim, which asks whether $\dim_{A,l}(G)$11, is NP-complete. The same source proves that it remains NP-complete on planar instances, and that, assuming ETH, there is no

$\dim_{A,l}(G)$12

algorithm solving LocAdjDim on graphs of order $\dim_{A,l}(G)$13 and size $\dim_{A,l}(G)$14. Through the corona identity

$\dim_{A,l}(G)$15

this hardness transfers to local metric dimension: LocDim is NP-complete and admits the same ETH-based lower bound. The complexity theory therefore matches the combinatorial richness of the parameter rather than reducing it to a simple neighborhood-cover problem (Rodríguez-Velázquez et al., 2013).

Two structural directions recur in the literature. One is equality theory. For the clique-number bound

$\dim_{A,l}(G)$16

infinitely many extremal graphs are known, but no complete characterization of all equality graphs is given. Another is the control of correction terms in product formulas: in lexicographic products, the distinction between

$\dim_{A,l}(G)$17

is encoded by $\dim_{A,l}(G)$18 and $\dim_{A,l}(G)$19, and equality between the two parameters is equivalent to the equality of those correction terms (Ghalavand et al., 18 Jul 2025, Barragán-Ramírez et al., 2016).

Taken together, these results place local adjacency metric dimension at a junction of local resolvability, product-graph structure, extremal graph theory, and complexity. The invariant is local in the sense that only edges matter, adjacency-based in the sense that longer distances are discarded, and sufficiently rigid to support sharp formulas, dichotomy theorems, and clique-number bounds across broad graph classes.