--- title: Local Multiset Dimension url: https://www.emergentmind.com/topics/local-multiset-dimension type: topic --- # Local Multiset Dimension 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 \(ldim_{ms}(G)\), only requires this distinction for adjacent vertices outside \(W\), and unlike \(lmd(G)\), it is always finite [2507.15071]. ## 1. Formal definition and notation Throughout, \(G\) is a finite, connected, undirected, and simple graph, with vertex set \(V(G)\) and edge set \(E(G)\). The graph distance between vertices \(u,w \in V(G)\) is denoted \(d_G(u,w)\), or \(d(u,w)\) for brevity. Given a set \(W \subseteq V(G)\), the representation multiset of \(u \in V(G)\) with respect to \(W\) is \[ M_W(u)=\ll d_G(u,w): w\in W\gg, \] written in the paper as \(m(u\mid W)\). 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, \[ m(u\mid W)=\ll 0^{a_0},1^{a_1},2^{a_2},\ldots\gg, \] where \(a_i\) is the multiplicity of the distance \(i\). A set \(W \subseteq V(G)\) is a local multiset resolving set if for all adjacent vertices \(u,v \in V(G)\), \[ m(u\mid W)\neq m(v\mid W). \] If such a set exists, a minimum-cardinality one is a local multiset basis, and its cardinality is the local multiset dimension \(lmd(G)\). If no such set exists, then \(lmd(G)=\infty\). A set \(W \subseteq V(G)\) is a local outer multiset resolving set if for all adjacent vertices \(u,v \in V(G)\setminus W\), \[ m(u\mid W)\neq m(v\mid W). \] A minimum-cardinality such set is a local outer multiset basis, and its cardinality is the local outer multiset dimension \(ldim_{ms}(G)\) [2507.15071]. ## 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 \[ r(u\mid W)=(d(u,w_1),\ldots,d(u,w_k)) \] for an ordered landmark set \(W=\{w_1,\ldots,w_k\}\), and requires distinctness for all vertex pairs. The local metric dimension \(ldim(G)\) 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 \(md(G)\), and sometimes \(lmd(G)\), to be infinite [2507.15071]. The paper records the following inequalities: \[ 1\le ldim(G)\le dim(G)\le dim_{ms}(G)\le md(G), \] \[ 1\le ldim_{ms}(G)\le dim_{ms}(G)\le n-1, \] \[ 1\le ldim(G)\le ldim_{ms}(G)\le lmd(G)\le md(G). \] Here \(dim(G)\) is the metric dimension, \(ldim(G)\) is the local metric dimension, \(dim_{ms}(G)\) is the outer multiset dimension, and \(md(G)\) is the multiset dimension. A fundamental distinction is that \(ldim_{ms}(G)\) is always finite, since \(ldim_{ms}(G)\le dim_{ms}(G)\le n-1\), whereas \(lmd(G)\) may be infinite [2507.15071]. The outer multiset dimension satisfies a sharp order bound: if \(|V(G)|=n\), then \(dim_{ms}(G)\le n-1\), and \(dim_{ms}(G)=n-1\) if and only if \(G\) is regular with diameter at most \(2\). This immediately yields finiteness of \(ldim_{ms}(G)\) for every graph [2507.15071]. A distinct use of the phrase “local multiset dimension” appears in poset theory. There, for a finite poset \(P\), the local dimension \(ldim(P)\) 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 [2001.08628]. This suggests a terminological distinction is necessary between the graph invariant \(lmd(G)\) and the poset invariant \(ldim(P)\). ## 3. Finiteness phenomena and structural obstructions A central structural obstruction is expressed through \(K_r\)-end vertices. For \(r\ge 3\), a vertex \(u\) in a clique \(K_r\) of \(G\) is a \(K_r\)-end vertex if \(\deg_G(u)=r-1\), that is, \(u\) has no neighbors outside the clique. If \(G\) has finite local multiset dimension, then every clique \(K\) of \(G\) contains at most two \(K\)-end vertices. Moreover, if a clique \(K\) contains exactly two \(K\)-end vertices, then every local multiset resolving set must include exactly one of them. The corresponding local outer statement is sharper: if a clique \(K\) contains more than two \(K\)-end vertices, then all of them, except one, belong to every local outer multiset resolving set of \(G\) [2507.15071]. The threshold value \(1\) is completely characterized. One has \[ lmd(G)=ldim_{ms}(G)=1 \quad\text{if and only if}\quad G \text{ is bipartite}. \] Equivalently, if \(G\) is not bipartite, then \[ lmd(G)\ge 2 \quad\text{and}\quad ldim_{ms}(G)\ge 2. \] 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 [2507.15071]. These finiteness phenomena contrast sharply with diameter-driven pathologies. The paper notes that graphs of diameter at most \(2\) have infinite multiset dimension \(md(G)\), 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 [2507.15071]. ## 4. Exact values for standard graph families Several graph classes admit complete evaluations. | Graph family | \(lmd(G)\) | \(ldim_{ms}(G)\) | |---|---:|---:| | \(K_n\) | \(1\) if \(n=2\); \(\infty\) if \(n\ge 3\) | \(n-1\) | | \(C_n\) | \(1\) if \(n\) even; \(3\) if \(n\ge 7\) odd; \(\infty\) if \(n=3,5\) | \(1\) if \(n\) even; \(2\) if \(n\) odd | | Non-cycle unicyclic graphs | \(1\) if the unique cycle is even; \(2\) if it is odd | same as \(lmd(G)\) | For complete graphs, the contrast between local and local outer behavior is extreme. One has \(lmd(K_n)=\infty\) for \(n\ge 3\), yet \(ldim_{ms}(K_n)=dim_{ms}(K_n)=n-1\). For cycles, even cycles behave bipartitely, odd cycles \(C_n\) with \(n\ge 7\) have \(lmd(C_n)=3\), and the small odd cycles \(C_3\) and \(C_5\) have \(lmd(C_n)=\infty\). For non-cycle unicyclic graphs, the value is determined entirely by the parity of the unique cycle [2507.15071]. Wheels \(W_n\), where a hub is joined to all vertices of a rim cycle \(W_n'\cong C_n\), exhibit a more intricate pattern. The paper proves \[ lmd(W_n)= \begin{cases} 3, & \text{if } n=4 \text{ or } 6,\\ \lceil n/4\rceil, & \text{if } n\ge 8 \text{ is even},\\ \infty, & \text{otherwise,} \end{cases} \] and \[ ldim_{ms}(W_n)= \begin{cases} 3, & \text{if } n=3,4,6,\\ \lceil n/4\rceil, & \text{if } n\ge 8 \text{ is even or } n\equiv 1 \pmod 4,\\ \lceil n/4\rceil+1, & \text{otherwise.} \end{cases} \] The proof relies on structural constraints for the induced rim subgraphs on \(W\) and on \(W_n\setminus W\): each can only contain rim-paths of orders \(1\) or \(3\) [2507.15071]. The paper also determines exact formulas for amalgamations of complete graphs. For the vertex amalgamation \(Amal(K_{n_i},m)\), finite local multiset dimension occurs if and only if \(n_i\le 3\) for all \(i\). If \(m_3\) is the number of \(K_3\)'s, then \[ lmd(Amal(K_{n_i},m))= \begin{cases} 1, & \text{if } m_3=0,\\ 2, & \text{if } m_3=1,\\ m_3, & \text{if } m_3\ge 2. \end{cases} \] If \(m_{\ge 4}\) is the number of \(K_n\) with \(n\ge 4\), then \[ ldim_{ms}(Amal(K_{n_i},m))= \begin{cases} 1, & \text{if } m_3=0 \text{ and } m_{\ge 4}=0,\\ 2, & \text{if } m_3=1 \text{ and } m_{\ge 4}=0,\\ \sum_{n_i\ge 3}(n_i-2), & \text{otherwise.} \end{cases} \] For the edge amalgamation \(EdgeAmal(K_{n_i},m)\), finite \(lmd\) occurs if and only if \(n_i\le 4\) for all \(i\). If \(m_2,m_3,m_4\) count \(K_2,K_3,K_4\), respectively, then \[ lmd(EdgeAmal(K_{n_i},m))= \begin{cases} 1, & \text{if } m_2=m,\\ 3, & \text{if } m_3\neq 0 \text{ and } m_2+m_3=m,\\ m_4+1, & \text{if } m_4\neq 0. \end{cases} \] If \(m_{\ge 4}\) counts the \(K_n\) with \(n\ge 4\), then \[ ldim_{ms}(EdgeAmal(K_{n_i},m))= \begin{cases} 1, & \text{if } m_2=m,\\ 2, & \text{if } 1\le m_3\le 2 \text{ and } m_2+m_3=m,\\ 3, & \text{if } m_3\ge 3 \text{ and } m_2+m_3=m,\\ \left(\sum_{n_i\ge 4}(n_i-3)\right)+1, & \text{if } m_{\ge 4}\neq 0. \end{cases} \] For corona products \(G\odot K_{m_i}\), the results are lower bounds and finiteness conditions rather than complete formulas. If \(lmd(G\odot K_{m_i})\) is finite, then \(m_i\le 2\) for all \(i\). Moreover, if \(m_i\le 2\), then \(lmd(G\odot K_{m_i})\ge m\), and this bound is sharp. Also, \[ ldim_{ms}(G\odot K_{m_i})\ge \sum_{i=1}^m (m_i-1), \] and this bound is sharp [2507.15071]. ## 5. Lower bounds, sharpness, and upper bounds Two general lower bounds organize much of the theory. The first is clique-based: if \(\omega(G)\) is the order of a maximal clique in \(G\), then \[ ldim_{ms}(G)\ge \lceil \log_2 \omega(G)\rceil \quad\text{and}\quad lmd(G)\ge \lceil \log_2 \omega(G)\rceil. \] This is sharp: for each \(n\ge 2\) there exists a graph \(G\) with \(\omega(G)=n\) such that \[ lmd(G)=ldim_{ms}(G)=\lceil \log_2 n\rceil. \] 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 [2507.15071]. The second bound couples diameter and chromatic number. If \(diam(G)=d\ge 2\), and \(g(d,\chi(G))\) is the least integer \(k\) such that \[ \binom{k+d-1}{d-1}+\binom{k+d-2}{d-1}-d+1 \ge \chi(G), \] then \[ lmd(G)\ge g(d,\chi(G)) \quad\text{and}\quad ldim_{ms}(G)\ge g(d,\chi(G)). \] These bounds are sharp for \(\chi(G)=2\), that is, for bipartite graphs. The resulting diameter-specific corollaries are \[ diam(G)=2 \implies lmd(G)\ge \chi(G)/2,\quad ldim_{ms}(G)\ge \chi(G)/2, \] and \[ diam(G)=3 \implies lmd(G)\ge \sqrt{\chi(G)+2}-1,\quad ldim_{ms}(G)\ge \sqrt{\chi(G)+2}-1. \] On the upper-bound side, if \(H\) is a maximal subgraph of \(G\) without leaves, then: \[ \text{if } lmd(H)\text{ is finite, then } lmd(G)\le lmd(H), \] and \[ ldim_{ms}(G)\le ldim_{ms}(H). \] These bounds are sharp; the paper gives \(G\cong H\odot K_2\) as an example where distances from leaves to landmarks in \(H\) are shifted uniformly by \(+1\), so a local or local outer multiset resolving set for \(H\) also resolves \(G\) in the corresponding sense [2507.15071]. ## 6. Constructions, examples, and open directions The sharpness construction for the clique lower bound is explicit. For \(n=2^k\), one takes a \(K_{2^k}\) whose vertices are partitioned into \(u_1,\ldots,u_k\) and \(v_0,\ldots,v_{2^k-k-1}\), attaches a path \(P_j\) of length \(2j\) to each \(u_j\), and connects each \(v_i\) to \(u_{j,1}\) according to an encoding \(a_{i,j}\in\{2j,2j+1\}\). Choosing \[ W=\{u_{1,2},u_{2,4},\ldots,u_{k,2k}\} \] produces \(k\)-bit local multiset patterns, and for \(2^{k-1}