Computing Metric Dimension of Two Types of Claw-free Cubic Graphs with Applications
Bibliographic record
Abstract
Consider the simple connected graph G with vertex set V(G) and edge set E(G). A graph \(G\) can be resolved by \(R\) if each vertex’s representation of distances to the other vertices in \(R\) uniquely identifies it. The minimum cardinality of the set \(R\) is the metric dimension of \(G\). The length of the shortest path between any two vertices, x, y in V(G), is signified by the distance symbol d(x, y). An ordered k-tuple \(r(x/R)=(d(x,z_1),d(x,\ z_2),…,d(x,z_k))\) represents representation of \(x\) with respect to \(R\) for an ordered subset \(R={\{z}_1,z_2,z_3…,z_k\}\) of vertices and vertex \(x\) in a connected graph. Metric dimension is used in a wide range of contexts where connection, distance, and connectedness are essential factors. It facilitates understanding the structure and dynamics of complex networks and problems relating to robotics network design, navigation, optimization, and facility location. Robots can optimize their localization and navigation methods using a small number of reference sites due to the pertinent idea of metric dimension. As a result, many robotic applications, such as collaborative robotics, autonomous navigation, and environment mapping, are more accurate, efficient, and resilient. A claw-free cubic graph (CCG) is one in which no induced subgraph is a claw. CCG proves helpful in various fields, including optimization, network design, and algorithm development. They offer intriguing structural and algorithmic properties. Developing algorithms and results for claw-free graphs frequently has applications in solving of challenging real-world situations. The metric dimension of a couple of claw-free cubic graphs (CCG), a string of diamonds (SOD), and a ring of diamonds (ROD) will be determined in this work.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".