Local Complement Metric Dimension of Sierpinski Gasket Graph and Hanoi Graph
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it