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 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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".