Local Multiset Dimension

• Local multiset dimension is a graph invariant defined using unordered distance multisets to distinguish adjacent vertices.
• It relies on selecting local multiset resolving sets that ensure adjacent vertices have distinct multiset profiles, with finite values in bipartite graphs.
• Exact formulas and bounds are established for standard graph families, revealing key contrasts with metric and outer multiset dimensions.

Searching arXiv for the cited papers to ground the article and verify the relevant terminology. Local multiset dimension is a graph invariant defined from unordered distance data. For a finite, connected, undirected, and simple graph $G$, and a vertex set $W \subseteq V(G)$, the representation multiset of a vertex $u$ with respect to $W$ is the multiset of distances from $u$ to the vertices of $W$. A set $W$ is a local multiset resolving set if every two adjacent vertices of $G$ have distinct multiset representations; the minimum cardinality of such a set is the local multiset dimension, denoted $lmd(G)$. If no such set exists, then $lmd(G)=\infty$. The associated local outer multiset dimension, denoted $W \subseteq V(G)$0, only requires this distinction for adjacent vertices outside $W \subseteq V(G)$1, and unlike $W \subseteq V(G)$2, it is always finite (Simanjuntak et al., 20 Jul 2025).

1. Formal definition and notation

Throughout, $W \subseteq V(G)$3 is a finite, connected, undirected, and simple graph, with vertex set $W \subseteq V(G)$4 and edge set $W \subseteq V(G)$5. The graph distance between vertices $W \subseteq V(G)$6 is denoted $W \subseteq V(G)$7, or $W \subseteq V(G)$8 for brevity.

Given a set $W \subseteq V(G)$9, the representation multiset of $u$0 with respect to $u$1 is

$u$2

written in the paper as $u$3. Because this is a multiset rather than an ordered tuple, multiplicities matter but positions do not. The same object may be encoded as a count profile by distance,

$u$4

where $u$5 is the multiplicity of the distance $u$6.

A set $u$7 is a local multiset resolving set if for all adjacent vertices $u$8,

$u$9

If such a set exists, a minimum-cardinality one is a local multiset basis, and its cardinality is the local multiset dimension $W$0. If no such set exists, then $W$1.

A set $W$2 is a local outer multiset resolving set if for all adjacent vertices $W$3,

$W$4

A minimum-cardinality such set is a local outer multiset basis, and its cardinality is the local outer multiset dimension $W$5 (Simanjuntak et al., 20 Jul 2025).

2. Relation to other dimensions and terminological scope

Local multiset dimension sits between several classical distance-based invariants. The classical metric dimension uses ordered distance vectors

$W$6

for an ordered landmark set $W$7, and requires distinctness for all vertex pairs. The local metric dimension $W$8 only requires distinctness for adjacent pairs. By contrast, the multiset framework discards the order of landmarks and keeps only the multiset of distances. This loss of order can reduce resolving power and can force the multiset dimension $W$9, and sometimes $u$0, to be infinite (Simanjuntak et al., 20 Jul 2025).

The paper records the following inequalities: $u$1

$u$2

$u$3

Here $u$4 is the metric dimension, $u$5 is the local metric dimension, $u$6 is the outer multiset dimension, and $u$7 is the multiset dimension. A fundamental distinction is that $u$8 is always finite, since $u$9, whereas $W$0 may be infinite (Simanjuntak et al., 20 Jul 2025).

The outer multiset dimension satisfies a sharp order bound: if $W$1, then $W$2, and $W$3 if and only if $W$4 is regular with diameter at most $W$5. This immediately yields finiteness of $W$6 for every graph (Simanjuntak et al., 20 Jul 2025).

A distinct use of the phrase “local multiset dimension” appears in poset theory. There, for a finite poset $W$7, the local dimension $W$8 is defined through local realizers and multiplicities of partial linear extensions, and the paper on Boolean-lattice suborders explicitly states that this parameter is what many authors call the local multiset dimension; in that setting, “local dimension” and “local multiset dimension” coincide (Lewis, 2020). This suggests a terminological distinction is necessary between the graph invariant $W$9 and the poset invariant $W$0.

3. Finiteness phenomena and structural obstructions

A central structural obstruction is expressed through $W$1-end vertices. For $W$2, a vertex $W$3 in a clique $W$4 of $W$5 is a $W$6-end vertex if $W$7, that is, $W$8 has no neighbors outside the clique. If $W$9 has finite local multiset dimension, then every clique $G$0 of $G$1 contains at most two $G$2-end vertices. Moreover, if a clique $G$3 contains exactly two $G$4-end vertices, then every local multiset resolving set must include exactly one of them. The corresponding local outer statement is sharper: if a clique $G$5 contains more than two $G$6-end vertices, then all of them, except one, belong to every local outer multiset resolving set of $G$7 (Simanjuntak et al., 20 Jul 2025).

The threshold value $G$8 is completely characterized. One has

$G$9

Equivalently, if $lmd(G)$0 is not bipartite, then

$lmd(G)$1

The parity-layer proof is intrinsic to the local nature of the parameter: with one landmark in one part of a bipartition, all adjacent vertices lie at distances of opposite parity (Simanjuntak et al., 20 Jul 2025).

These finiteness phenomena contrast sharply with diameter-driven pathologies. The paper notes that graphs of diameter at most $lmd(G)$2 have infinite multiset dimension $lmd(G)$3, while the local outer multiset dimension can still be finite and even comparatively small. A plausible implication is that locality partly mitigates, but does not eliminate, the symmetry collapse caused by passing from ordered distance vectors to unordered distance multisets (Simanjuntak et al., 20 Jul 2025).

4. Exact values for standard graph families

Several graph classes admit complete evaluations.

Graph family	$lmd(G)$4	$lmd(G)$5
$lmd(G)$6	$lmd(G)$7 if $lmd(G)$8; $lmd(G)$9 if $lmd(G)=\infty$0	$lmd(G)=\infty$1
$lmd(G)=\infty$2	$lmd(G)=\infty$3 if $lmd(G)=\infty$4 even; $lmd(G)=\infty$5 if $lmd(G)=\infty$6 odd; $lmd(G)=\infty$7 if $lmd(G)=\infty$8	$lmd(G)=\infty$9 if $W \subseteq V(G)$00 even; $W \subseteq V(G)$01 if $W \subseteq V(G)$02 odd
Non-cycle unicyclic graphs	$W \subseteq V(G)$03 if the unique cycle is even; $W \subseteq V(G)$04 if it is odd	same as $W \subseteq V(G)$05

For complete graphs, the contrast between local and local outer behavior is extreme. One has $W \subseteq V(G)$06 for $W \subseteq V(G)$07, yet $W \subseteq V(G)$08. For cycles, even cycles behave bipartitely, odd cycles $W \subseteq V(G)$09 with $W \subseteq V(G)$10 have $W \subseteq V(G)$11, and the small odd cycles $W \subseteq V(G)$12 and $W \subseteq V(G)$13 have $W \subseteq V(G)$14. For non-cycle unicyclic graphs, the value is determined entirely by the parity of the unique cycle (Simanjuntak et al., 20 Jul 2025).

Wheels $W \subseteq V(G)$15, where a hub is joined to all vertices of a rim cycle $W \subseteq V(G)$16, exhibit a more intricate pattern. The paper proves

$W \subseteq V(G)$17

and

$W \subseteq V(G)$18

The proof relies on structural constraints for the induced rim subgraphs on $W \subseteq V(G)$19 and on $W \subseteq V(G)$20: each can only contain rim-paths of orders $W \subseteq V(G)$21 or $W \subseteq V(G)$22 (Simanjuntak et al., 20 Jul 2025).

The paper also determines exact formulas for amalgamations of complete graphs. For the vertex amalgamation $W \subseteq V(G)$23, finite local multiset dimension occurs if and only if $W \subseteq V(G)$24 for all $W \subseteq V(G)$25. If $W \subseteq V(G)$26 is the number of $W \subseteq V(G)$27's, then

$W \subseteq V(G)$28

If $W \subseteq V(G)$29 is the number of $W \subseteq V(G)$30 with $W \subseteq V(G)$31, then

$W \subseteq V(G)$32

For the edge amalgamation $W \subseteq V(G)$33, finite $W \subseteq V(G)$34 occurs if and only if $W \subseteq V(G)$35 for all $W \subseteq V(G)$36. If $W \subseteq V(G)$37 count $W \subseteq V(G)$38, respectively, then

$W \subseteq V(G)$39

If $W \subseteq V(G)$40 counts the $W \subseteq V(G)$41 with $W \subseteq V(G)$42, then

$W \subseteq V(G)$43

For corona products $W \subseteq V(G)$44, the results are lower bounds and finiteness conditions rather than complete formulas. If $W \subseteq V(G)$45 is finite, then $W \subseteq V(G)$46 for all $W \subseteq V(G)$47. Moreover, if $W \subseteq V(G)$48, then $W \subseteq V(G)$49, and this bound is sharp. Also,

$W \subseteq V(G)$50

and this bound is sharp (Simanjuntak et al., 20 Jul 2025).

5. Lower bounds, sharpness, and upper bounds

Two general lower bounds organize much of the theory. The first is clique-based: if $W \subseteq V(G)$51 is the order of a maximal clique in $W \subseteq V(G)$52, then

$W \subseteq V(G)$53

This is sharp: for each $W \subseteq V(G)$54 there exists a graph $W \subseteq V(G)$55 with $W \subseteq V(G)$56 such that

$W \subseteq V(G)$57

The counting argument is based on the fact that a clique vertex can realize at most two distance possibilities per landmark in the multiset profile (Simanjuntak et al., 20 Jul 2025).

The second bound couples diameter and chromatic number. If $W \subseteq V(G)$58, and $W \subseteq V(G)$59 is the least integer $W \subseteq V(G)$60 such that

$W \subseteq V(G)$61

then

$W \subseteq V(G)$62

These bounds are sharp for $W \subseteq V(G)$63, that is, for bipartite graphs. The resulting diameter-specific corollaries are

$W \subseteq V(G)$64

and

$W \subseteq V(G)$65

On the upper-bound side, if $W \subseteq V(G)$66 is a maximal subgraph of $W \subseteq V(G)$67 without leaves, then: $W \subseteq V(G)$68 and

$W \subseteq V(G)$69

These bounds are sharp; the paper gives $W \subseteq V(G)$70 as an example where distances from leaves to landmarks in $W \subseteq V(G)$71 are shifted uniformly by $W \subseteq V(G)$72, so a local or local outer multiset resolving set for $W \subseteq V(G)$73 also resolves $W \subseteq V(G)$74 in the corresponding sense (Simanjuntak et al., 20 Jul 2025).

6. Constructions, examples, and open directions

The sharpness construction for the clique lower bound is explicit. For $W \subseteq V(G)$75, one takes a $W \subseteq V(G)$76 whose vertices are partitioned into $W \subseteq V(G)$77 and $W \subseteq V(G)$78, attaches a path $W \subseteq V(G)$79 of length $W \subseteq V(G)$80 to each $W \subseteq V(G)$81, and connects each $W \subseteq V(G)$82 to $W \subseteq V(G)$83 according to an encoding $W \subseteq V(G)$84. Choosing

$W \subseteq V(G)$85

produces $W \subseteq V(G)$86-bit local multiset patterns, and for $W \subseteq V(G)$87, some $W \subseteq V(G)$88's are deleted (Simanjuntak et al., 20 Jul 2025).

For wheels, the proof method is a path-tiling argument on the rim. The induced subgraphs $W \subseteq V(G)$89 and $W \subseteq V(G)$90 can only contain rim-paths of orders $W \subseteq V(G)$91 or $W \subseteq V(G)$92. Alternating these short paths yields the exact formulas for $W \subseteq V(G)$93 and $W \subseteq V(G)$94. This suggests that local multiset resolvability on small-diameter graphs is strongly constrained by short path decompositions, although the paper formulates this only for wheels (Simanjuntak et al., 20 Jul 2025).

The paper does not provide complexity results, algorithms, or hardness proofs for computing $W \subseteq V(G)$95 or $W \subseteq V(G)$96. Its emphasis is structural: lower bounds, necessary conditions for finiteness, and exact values for specific families (Simanjuntak et al., 20 Jul 2025).

Several worked examples illustrate the definitions. Since paths are bipartite, $W \subseteq V(G)$97. For the cycle $W \subseteq V(G)$98, one has $W \subseteq V(G)$99 and $u$00; taking two adjacent landmarks on the cycle distinguishes all adjacent vertices outside the landmark set. For $u$01, $u$02 while $u$03. For the star $u$04, bipartiteness again gives $u$05. For the wheel $u$06, the exact value is $u$07 and $u$08 (Simanjuntak et al., 20 Jul 2025).

The paper isolates several open problems. One asks whether, for any $u$09, there exists a graph $u$10 with $u$11 such that $u$12, and similarly for $u$13. Another asks for good upper bounds for $u$14 and $u$15. A further direction concerns graph joins $u$16: since joins typically have diameter at most $u$17, the paper asks for formulas for $u$18 and $u$19 in terms of the corresponding parameters of $u$20 and $u$21 (Simanjuntak et al., 20 Jul 2025).