Multiset Metric Dimension in Graphs

• Multiset metric dimension is defined as the minimum size of a landmark set whose distance multiset uniquely identifies every vertex in a graph, highlighting its rigorous distinguishing power.
• It leverages distance multisets rather than ordered vectors, leading to key differences from classical metric dimension, with precise results established for trees, cycles, and grids.
• The approach has broad applications in probabilistic graph theory and chemical graph analysis, while its NP-completeness and outer variant offer insights into computational boundaries.

Multiset metric dimension, often called multiset dimension, is a distance-based graph invariant in which vertices are identified by the multiset of their distances to a chosen landmark set rather than by an ordered distance vector. For a connected graph $G$, a landmark set is multiset resolving when distinct vertices induce distinct distance multisets; the minimum size of such a set is the multiset metric dimension, and it is defined to be $\infty$ when no such set exists. Introduced by Simanjuntak, Siagian, and Vetrík in 2017, the parameter is closely related to classical metric dimension but is substantially more rigid: the loss of positional information can force infinitude even for connected graphs, especially in diameter-$2$ and highly symmetric settings (Simanjuntak et al., 2017, Eide et al., 15 Jul 2025).

1. Definition and formal representations

Let $G=(V,E)$ be a connected graph and let $W\subseteq V(G)$. The original formulation defines the representation multiset of a vertex $v$ with respect to $W$ as $r_m(v\mid W)$, the multiset of distances from $v$ to the vertices of $W$. A set $\infty$0 is an $\infty$1-resolving set if $\infty$2 for every pair of distinct vertices $\infty$3, and the minimum cardinality of such a set is $\infty$4; if no such set exists, then $\infty$5 (Simanjuntak et al., 2017).

Several equivalent notational frameworks appear in the literature. For $\infty$6, one may define

$\infty$7

so that $\infty$8 records the multiplicity of each distance value and is equivalent to the multiset $\infty$9. In another notation, $2$0, while chemically motivated work uses $2$1 for the same object. These formulations are interchangeable: a set is multiset resolving exactly when these multiset or count-based signatures are pairwise distinct (Eide et al., 15 Jul 2025, Hakanen et al., 2023, Liu et al., 2021).

The distinction from ordinary metric dimension is fundamental. In the classical setting one fixes an ordered landmark set and records a distance vector $2$2; in the multiset setting, order is discarded and only multiplicities are retained. Consequently, different metric representations may collapse to the same multiset representation under permutation of coordinates. Every multiset resolving set is therefore a resolving set in the ordinary sense, yielding the basic inequality

$2$3

or equivalently $2$4 in later notation (Simanjuntak et al., 2017, Hakanen et al., 2023).

2. Foundational properties and obstructions to finiteness

The baseline theory is unusually rigid. The graphs of multiset dimension $2$5 are exactly the paths, and no graph has multiset dimension $2$6. Hence every non-path graph satisfies $2$7 (Simanjuntak et al., 2017). This already separates the parameter sharply from classical metric dimension, where value $2$8 is common.

A central obstruction is small diameter. If $2$9 is a non-path graph with diameter at most $G=(V,E)$0, then $G=(V,E)$1. In particular, every non-path graph of diameter $G=(V,E)$2 has no multiset resolving set. This explains why complete graphs, stars, cycles $G=(V,E)$3 for $G=(V,E)$4, the Petersen graph, and strongly regular graphs fall into the infinite regime, and it underlies the observation that almost all graphs have infinite multiset dimension (Simanjuntak et al., 2017, Hafidh et al., 2019, Eide et al., 15 Jul 2025).

Twin structure provides a second major obstruction. If some twin class $G=(V,E)$5 has size at least $G=(V,E)$6, then $G=(V,E)$7, because three or more vertices with identical distances to every other vertex cannot all be separated by distance multisets. Moreover, if $G=(V,E)$8 and $G=(V,E)$9, then every $W\subseteq V(G)$0-resolving set must contain exactly one vertex from that twin pair. These conditions are necessary but not sufficient: there are trees of diameter $W\subseteq V(G)$1 with no twin class of size at least $W\subseteq V(G)$2 that still have infinite multiset dimension, showing that more subtle local symmetries also matter (Simanjuntak et al., 2017).

The original theory also gives a counting lower bound in terms of order and diameter. If $W\subseteq V(G)$3 has order $W\subseteq V(G)$4 and diameter $W\subseteq V(G)$5, and $W\subseteq V(G)$6 is the least positive integer $W\subseteq V(G)$7 such that

$W\subseteq V(G)$8

then $W\subseteq V(G)$9. The bound reflects the number of multisets of size $v$0 with entries from $v$1 available to represent vertices outside the landmark set (Simanjuntak et al., 2017).

3. Exact results for trees, cycles, and grids

Trees occupy a central place because they exhibit both finiteness and obstruction phenomena. For a tree $v$2 of order $v$3 and diameter at least $v$4, if $v$5, then

$v$6

This improves the trivial $v$7-vertex upper bound for trees with finite multiset dimension. The same work proposes the sharper conjecture that, if $v$8, then

$v$9

and reports exhaustive computer search for all trees up to order $W$0 in support of that conjecture. The paper also characterizes two tree subclasses completely: a caterpillar has finite multiset dimension if and only if every vertex of its minimum $W$1-center path has at most $W$2 neighbors in $W$3; a lobster has finite multiset dimension if and only if the only component of $W$4 with infinite multiset dimension is an $W$5 (Hafidh et al., 2019).

Among exact families, the original paper shows that complete $W$6-ary trees have finite multiset dimension if and only if $W$7 or $W$8. For a complete binary tree of height $W$9,

$r_m(v\mid W)$0

It also establishes that

$r_m(v\mid W)$1

and

$r_m(v\mid W)$2

demonstrating that multiset dimension $r_m(v\mid W)$3 is attained by natural sparse families despite the nonexistence of value $r_m(v\mid W)$4 (Simanjuntak et al., 2017).

Chemically motivated graph classes supply further exact computations. For the starphene family $r_m(v\mid W)$5 with $r_m(v\mid W)$6,

$r_m(v\mid W)$7

The proof is constructive: a $r_m(v\mid W)$8-vertex multiresolving set is exhibited and explicit multirepresentation formulas are derived for all vertex classes. The same source notes a typographical inconsistency in the proof text, whose theorem statement and displayed generator indicate that the intended value is $r_m(v\mid W)$9, not $v$0 (Liu et al., 2021).

4. Equivalence, complexity, and product phenomena

A major conceptual advance is the equivalence between multiset resolving sets and ID-colorings. If $v$1 has diameter $v$2 and $v$3, one may encode each vertex $v$4 by the vector

$v$5

where $v$6 is the number of vertices of $v$7 at distance $v$8 from $v$9. The equivalence theorem states that $W$0 is an ID-coloring if and only if it is a multiset resolving set; consequently,

$W$1

This identifies multiset dimension with a count-vector formalism that is often more convenient for structural arguments (Hakanen et al., 2023).

The same paper proves that the decision problem MULTISET DIMENSION is NP-complete. Given a graph $W$2 and an integer $W$3, the question whether $W$4 is shown NP-complete by reduction from $W$5-SAT using variable and clause gadgets, twin-forcing arguments, and a tight target size $W$6 for a formula with $W$7 variables and $W$8 clauses (Hakanen et al., 2023).

Strong products display a marked dichotomy. For king grids $W$9, the parameter is infinite for $\infty$00 because these graphs have diameter at most $\infty$01; exhaustive search gives

$\infty$02

and for $\infty$03,

$\infty$04

Whether the upper bound $\infty$05 is always tight for $\infty$06 is left open. For products with a complete graph factor, finite multiset dimension occurs only in a highly restricted case: $\infty$07 When this holds,

$\infty$08

These results show that product operations can preserve or destroy finiteness depending on the twin structure induced in the fibers (Hakanen et al., 2023).

5. Outer multiset dimension and related variants

Because full multiset dimension is often infinite, an important variant is the outer multiset dimension. Here one chooses $\infty$09 and requires only that vertices in $\infty$10 have pairwise distinct multiset representations with respect to $\infty$11. This weakens the recognition requirement by allowing vertices inside $\infty$12 to remain unresolved, thereby avoiding many of the pathologies of the full parameter (Klavzar et al., 2022).

The outer version admits structural and algorithmic results not presently available in the same form for the full invariant. A central theorem states

$\infty$13

Graphs with outer multiset dimension $\infty$14 are recognized in polynomial time; specifically, the decision can be done in $\infty$15 time. The paper also proves that for $\infty$16,

$\infty$17

with equality if and only if $\infty$18 is multiset distance irregular, and establishes the exact Cartesian-grid value

$\infty$19

This suggests that the outer version is both structurally rich and computationally more tractable in some regimes (Klavzar et al., 2022).

Another related notion is mixed metric dimension, where a landmark set must distinguish both vertices and edges. The starphene/coronoid study explicitly separates mixed metric dimension from multiset dimension: multiset dimension is purely vertex-based and uses multiset-valued distance lists, whereas mixed metric dimension resolves all elements of $\infty$20. For starphene,

$\infty$21

while prior values cited there give

$\infty$22

Thus the multiset requirement can be strictly stronger than both ordinary and mixed distance-based identification on the same graph family (Liu et al., 2021).

6. Random graphs and asymptotic phase behavior

The first high-probability asymptotic bounds for the multiset metric dimension of binomial random graphs are obtained for $\infty$23 in the regime

$\infty$24

For $\infty$25, the relevant signature is

$\infty$26

and the analysis is driven by an explicit exponent function

$\infty$27

When $\infty$28, $\infty$29 is defined as the unique solution of $\infty$30, and when $\infty$31, $\infty$32 is defined as the unique solution of $\infty$33 (Eide et al., 15 Jul 2025).

The main theorem gives high-probability polynomial bounds. If

$\infty$34

then with high probability

$\infty$35

Equivalently, for $\infty$36 with $\infty$37,

$\infty$38

If

$\infty$39

then with high probability

$\infty$40

equivalently

$\infty$41

For $\infty$42, $\infty$43 has diameter $\infty$44 with high probability, and therefore

$\infty$45

by the general diameter-$\infty$46 obstruction (Eide et al., 15 Jul 2025).

The proof combines expansion properties of $\infty$47 in the regime $\infty$48 with probabilistic control of distance layers. A key lemma gives typical sphere sizes

$\infty$49

where $\infty$50 is the largest integer with $\infty$51. For the upper bound, each vertex is chosen independently with probability $\infty$52, where $\infty$53, and the probability that a random candidate set fails to distinguish a fixed pair is roughly

$\infty$54

For the lower bound, if $\infty$55, then typical vertices admit only

$\infty$56

possible values in coordinate $\infty$57, so the total number of typical signatures is at most

$\infty$58

The resulting phase picture has a qualitative transition: for $\infty$59 the parameter is infinite with high probability; for $\infty$60 polynomial lower bounds are proved; and for $\infty$61 polynomial upper bounds are also proved. The interval

$\infty$62

remains unresolved with respect to finiteness with high probability. The exponent functions $\infty$63 and $\infty$64 exhibit a “zig-zag” pattern with jumps at reciprocals $\infty$65, reflecting diameter changes in $\infty$66; examples given are

$\infty$67

These results place multiset metric dimension within probabilistic graph theory and show that, when finite in sparse polynomial-degree random graphs, it is typically polynomially large (Eide et al., 15 Jul 2025).