Local Outer Multiset Dimension in Graphs

• Local outer multiset dimension is defined for a finite, connected graph where a landmark set ensures that adjacent non-landmark vertices have distinct unordered distance multisets.
• The framework extends classic metric dimensions by using multisets instead of vectors, and it provides exact formulas and bounds for families like cycles, wheels, and complete graphs.
• Structural constraints such as clique end-vertex conditions lead to sharp lower bounds and open questions, highlighting practical implications for graph identification and complexity.

Searching arXiv for the specified papers to ground the article in current sources. arxiv_search(query="Local (Outer) Multiset Dimensions of Graphs", max_results=5) arxiv_search(query="Distance-based vertex identification in graphs: the outer multiset dimension", max_results=5) arxiv_search(query="(Simanjuntak et al., 20 Jul 2025)", max_results=5) Local outer multiset dimension is a distance-based graph parameter defined for a finite, connected, undirected, simple graph $G$. For a set $W \subseteq V(G)$, the representation multiset of a vertex $u$ with respect to $W$ is

$m(u|W)=\{d(u,w): w\in W\},$

viewed as a multiset rather than an ordered vector. The set $W$ is a local outer multiset resolving set if every two adjacent vertices in $V(G)\setminus W$ have distinct representation multisets, and the minimum cardinality of such a set is the local outer multiset dimension, denoted $\operatorname{ldim}_{ms}(G)$ (Simanjuntak et al., 20 Jul 2025). The notion is the local counterpart of the outer multiset dimension introduced earlier for global separation of all vertices outside the landmark set (Gil-Pons et al., 2019).

1. Formal framework and historical placement

The underlying distance model is inherited from multiset-based vertex identification. If $W=\{w_1,\dots,w_t\}\subseteq V(G)$, then $m(u|W)$ records the distances from $W \subseteq V(G)$0 to the vertices of $W \subseteq V(G)$1, but ignores the order of the landmarks. This distinguishes the theory from metric-dimension variants based on ordered distance vectors.

In the 2025 formulation, two local parameters are separated. A local multiset resolving set requires $W \subseteq V(G)$2 for every edge $W \subseteq V(G)$3. A local outer multiset resolving set weakens this by requiring the inequality only when both $W \subseteq V(G)$4 and $W \subseteq V(G)$5 lie outside $W \subseteq V(G)$6. The corresponding minimum cardinalities are the local multiset dimension $W \subseteq V(G)$7 and the local outer multiset dimension $W \subseteq V(G)$8, respectively (Simanjuntak et al., 20 Jul 2025).

Historically, the precursor is the outer multiset dimension studied in 2019. There, a set $W \subseteq V(G)$9 is required to distinguish all distinct vertices in $u$0, not merely adjacent ones. The 2019 paper explicitly does not define or analyze a local outer multiset dimension; all of its results concern the global requirement that all pairs in $u$1 be distinguished (Gil-Pons et al., 2019). The later local theory therefore extends the multiset-based framework by replacing global separation with adjacency-restricted separation.

2. Hierarchy among distance dimensions

The local outer multiset dimension sits between several established parameters. The inequalities

$u$2

and

$u$3

hold for every graph of order $u$4 (Simanjuntak et al., 20 Jul 2025). In addition, the broader chain

$u$5

links the local and global metric and multiset parameters.

These inequalities encode three important comparisons. First, replacing ordered distance vectors by multisets does not make local identification easier than local metric resolution, since $u$6. Second, restricting attention to adjacent pairs outside the landmark set makes the local outer requirement weaker than the global outer requirement, since $u$7. Third, the local multiset dimension may be strictly harder than the local outer version, since $u$8, and $u$9 may even be infinite (Simanjuntak et al., 20 Jul 2025).

A central finiteness property distinguishes the local outer parameter from the non-outer local multiset dimension. Every graph has finite local outer multiset dimension: if $W$0 and $W$1, then $W$2, so there is no adjacent pair outside $W$3, and the local outer condition holds vacuously. Hence $W$4 for every graph (Simanjuntak et al., 20 Jul 2025). This parallels the global outer theory, where the outer multiset dimension is always defined because taking the entire vertex set as landmarks is trivially outer resolving (Gil-Pons et al., 2019).

3. Structural constraints and general bounds

A recurring structural mechanism is the presence of clique end-vertices. For $W$5, a vertex $W$6 in a clique $W$7 is a $W$8-end vertex if $W$9. The necessary condition for finiteness of $m(u|W)=\{d(u,w): w\in W\},$0 states that if $m(u|W)=\{d(u,w): w\in W\},$1, then every clique contains at most two $m(u|W)=\{d(u,w): w\in W\},$2-end vertices; moreover, if a clique contains two $m(u|W)=\{d(u,w): w\in W\},$3-end vertices, then exactly one of them must belong to every local multiset resolving set. For the local outer parameter, if a clique contains more than two $m(u|W)=\{d(u,w): w\in W\},$4-end vertices, then all of them except one belong to every local outer multiset resolving set (Simanjuntak et al., 20 Jul 2025). This supplies a direct lower-bound mechanism in dense local structures.

The minimum possible value is characterized exactly: $m(u|W)=\{d(u,w): w\in W\},$5 if and only if $m(u|W)=\{d(u,w): w\in W\},$6 is bipartite. Consequently, any non-bipartite graph satisfies $m(u|W)=\{d(u,w): w\in W\},$7 and $m(u|W)=\{d(u,w): w\in W\},$8 (Simanjuntak et al., 20 Jul 2025). This is one of the sharpest available characterizations in the theory, because it identifies a purely structural class realizing the absolute minimum.

Two general lower bounds tie $m(u|W)=\{d(u,w): w\in W\},$9 to standard graph invariants. If $W$0 is the clique number, then

$W$1

The bound is sharp: for each $W$2, there exists a graph $W$3 with $W$4 such that $W$5 (Simanjuntak et al., 20 Jul 2025). A second bound uses diameter and chromatic number. If $W$6 and $W$7 is the chromatic number, let $W$8 be the smallest integer $W$9 such that

$V(G)\setminus W$0

Then

$V(G)\setminus W$1

For diameter $V(G)\setminus W$2, this yields $V(G)\setminus W$3; for diameter $V(G)\setminus W$4, it yields $V(G)\setminus W$5 (Simanjuntak et al., 20 Jul 2025).

There is also a useful monotonicity-style upper bound under leaf attachment. If $V(G)\setminus W$6 is a maximal subgraph of $V(G)\setminus W$7 with no leaf, then

$V(G)\setminus W$8

The bound is sharp, for example when $V(G)\setminus W$9 (Simanjuntak et al., 20 Jul 2025). This suggests that, for the local outer parameter, pendant growth does not increase complexity beyond that already present in the leafless core.

4. Exact values for basic graph families

Several standard graph classes admit complete formulas. For complete graphs,

$\operatorname{ldim}_{ms}(G)$0

This coincides with the global outer multiset value for $\operatorname{ldim}_{ms}(G)$1. By contrast, the local multiset dimension behaves differently: $\operatorname{ldim}_{ms}(G)$2 and $\operatorname{ldim}_{ms}(G)$3 for $\operatorname{ldim}_{ms}(G)$4 (Simanjuntak et al., 20 Jul 2025). Thus complete graphs simultaneously witness finiteness of the local outer parameter and infinitude of the non-outer local multiset parameter.

Cycles exhibit a sharp parity dichotomy. The local outer multiset dimension satisfies

$\operatorname{ldim}_{ms}(G)$5

For local multiset dimension, the corresponding formula is

$\operatorname{ldim}_{ms}(G)$6

For odd cycles, the paper uses adjacent landmarks $\operatorname{ldim}_{ms}(G)$7 and analyzes the longer $\operatorname{ldim}_{ms}(G)$8-$\operatorname{ldim}_{ms}(G)$9 path to show that all adjacent pairs outside $W=\{w_1,\dots,w_t\}\subseteq V(G)$0 receive distinct multisets (Simanjuntak et al., 20 Jul 2025).

Unicyclic graphs that are not cycles also admit a complete dichotomy. If such a graph contains an even cycle, then it is bipartite and

$W=\{w_1,\dots,w_t\}\subseteq V(G)$1

If it contains an odd cycle and is not itself a cycle, then

$W=\{w_1,\dots,w_t\}\subseteq V(G)$2

The proofs are constructive, using a 2-vertex set $W=\{w_1,\dots,w_t\}\subseteq V(G)$3 that resolves adjacent pairs by multiset distances (Simanjuntak et al., 20 Jul 2025).

These formulas show that locality drastically lowers the required number of landmarks in many sparse families. In particular, for cycles the local outer parameter is strictly smaller than the global outer multiset dimension studied earlier, where $W=\{w_1,\dots,w_t\}\subseteq V(G)$4, $W=\{w_1,\dots,w_t\}\subseteq V(G)$5, $W=\{w_1,\dots,w_t\}\subseteq V(G)$6, and $W=\{w_1,\dots,w_t\}\subseteq V(G)$7 for all $W=\{w_1,\dots,w_t\}\subseteq V(G)$8 (Gil-Pons et al., 2019).

5. Wheels, amalgamations, and corona products

For wheel graphs $W=\{w_1,\dots,w_t\}\subseteq V(G)$9 of diameter $m(u|W)$0, exact formulas are known. If $m(u|W)$1 is the hub and $m(u|W)$2 is the rim cycle, then

$m(u|W)$3

whereas

$m(u|W)$4

The key combinatorial restriction is that in the induced rim subgraphs $m(u|W)$5 and $m(u|W)$6, any induced path must have order $m(u|W)$7 or $m(u|W)$8. This forces the optimal periodic selection pattern on the rim (Simanjuntak et al., 20 Jul 2025).

Amalgamations of complete graphs provide further exact formulas. For vertex amalgamation $m(u|W)$9, with $W \subseteq V(G)$00 the number of $W \subseteq V(G)$01 components and $W \subseteq V(G)$02 the number of $W \subseteq V(G)$03 with $W \subseteq V(G)$04,

$W \subseteq V(G)$05

For edge amalgamation $W \subseteq V(G)$06, with $W \subseteq V(G)$07 counting the complete graphs of the indicated sizes,

$W \subseteq V(G)$08

In both constructions, the formulas are driven by the $W \subseteq V(G)$09-end vertex constraints, and the extra $W \subseteq V(G)$10 in edge amalgamation frequently arises from the need to include one of the identified vertices to separate remaining adjacent pairs outside the landmark set (Simanjuntak et al., 20 Jul 2025).

For corona products $W \subseteq V(G)$11, the available result is a sharp lower bound: $W \subseteq V(G)$12 The bound is sharp. It follows directly from the clique end-vertex constraints, since each attached clique forces the inclusion of all but one of its end-vertices in any local outer multiset resolving set (Simanjuntak et al., 20 Jul 2025).

6. Relation to global outer multiset dimension, extremal behavior, and open questions

The local outer multiset dimension is best understood against the background of the global outer multiset dimension. In the global version, a set $W \subseteq V(G)$13 must distinguish every pair of distinct vertices in $W \subseteq V(G)$14 through their multiset representations. The 2019 theory established the general bounds

$W \subseteq V(G)$15

proved exact values for paths, cycles, complete graphs, balanced complete multipartite graphs, and some wheels, and showed that deciding whether $W \subseteq V(G)$16 is NP-complete (Gil-Pons et al., 2019). By contrast, the 2025 local theory gives no explicit complexity classification and concentrates instead on structural bounds, exact formulas for families, and constructive proofs (Simanjuntak et al., 20 Jul 2025).

Several examples illustrate the gap between local and global outer requirements. For complete graphs, the two parameters coincide at the maximal value $W \subseteq V(G)$17. For cycles, however, the local outer values are $W \subseteq V(G)$18 or $W \subseteq V(G)$19, whereas the global outer values are $W \subseteq V(G)$20, $W \subseteq V(G)$21, or $W \subseteq V(G)$22 depending on $W \subseteq V(G)$23. This indicates that adjacency-restricted separation can be much weaker than global separation, even in highly symmetric graphs.

The theory also highlights extremal behavior. The value $W \subseteq V(G)$24 occurs exactly for bipartite graphs, so the minimum is completely classified. At the other extreme, $W \subseteq V(G)$25 can reach $W \subseteq V(G)$26, as shown by $W \subseteq V(G)$27. The clique-number lower bound $W \subseteq V(G)$28 is sharp, and the diameter/chromatic lower bound is sharp for $W \subseteq V(G)$29 (Simanjuntak et al., 20 Jul 2025).

Several open problems remain. For corona products $W \subseteq V(G)$30, good upper bounds for $W \subseteq V(G)$31 and $W \subseteq V(G)$32 are not known in general. For the diameter/chromatic lower bound, sharpness is open for $W \subseteq V(G)$33. For graph joins $W \subseteq V(G)$34, determining $W \subseteq V(G)$35 and $W \subseteq V(G)$36 in terms of the corresponding parameters of $W \subseteq V(G)$37 and $W \subseteq V(G)$38 is also open (Simanjuntak et al., 20 Jul 2025). Together, these questions indicate that the local outer multiset dimension is structurally well-founded but still far from exhausted, especially in dense diameter-$W \subseteq V(G)$39 regimes and in graph operations that amplify clique interactions.