Minimum Sizes of Identifying Codes in Graphs Differing by One Edge or One Vertex
Bibliographic record
Abstract
Let $G$ be a simple, undirected graph with vertex set $V$. For $v \in V$ and $r \geq 1$, we denote by $B_{G,r}(v)$ the ball of radius $r$ and centre $v$. A set $C \subseteq V$ is said to be an $r$-identifying code in $G$ if the sets $B_{G,r}(v) \cap C$, $v \in V$ , are all nonempty and distinct. A graph $G$ admitting an $r$- identifying code is called $r$-twin-free, and in this case the size of a smallest $r$-identifying code in $G$ is denoted by $\gamma_r(G)$. We study the following structural problem: let $G$ be an $r$-twin-free graph, and $G^*$ be a graph obtained from $G$ by adding or deleting a vertex, or by adding or deleting an edge. If $G^*$ is still $r$-twin-free, we compare the behaviours of $\gamma_r(G)$ and $\gamma_r(G^*)$, establishing results on their possible differences and ratios.
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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 | 0.002 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".