Local Complement Metric Dimension of Sierpinski Gasket Graph and Hanoi Graph
Bibliographic record
Abstract
One of the topics in graph theory that has attracted the attention of many researchers is the study of metric dimension. The metric dimension is a key concept in graph theory with wide-ranging applications in areas such as optimization, image processing, routing, and biological analysis. The metric dimension refers to finding the minimum set of marker vertices that uniquely distinguish every vertex in a graph from one another. For a connected graph $G$, a nonempty set $W \subseteq V(G)$ is a local complement resolving set if there exist adjacent vertices $u,v \in V(G)$ such that $r(u \mid W) = r(v \mid W)$. The largest such set is the local complement basis, and its size is the local complement metric dimension denoted by $\overline{\dim_l}(S_n)$. In this paper, we focus on the local complement metric dimension of two families of recursively defined graphs: the Sierpinski Gasket graph $S_n$ and the Hanoi graph $H_n$. We prove that for $n \geq 3$, $\overline{\dim_l}(S_n) = 2 + \sum_{k=3}^{n} \lvert V(S_k^2) \rvert$, and for $n \geq 2$, $\overline{\dim_l}(H_n) = 2 + \sum_{k=2}^{n} \lvert V(H_k^2) \rvert$. These results indicate that the recursive structures of the Sierpinski Gasket and Hanoi graphs play a crucial role in determining their local complement metric dimensions.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".