{"id":"57ec1c81-dff4-4a52-ab5b-09ecafc06e84","arxiv_id":"2607.05152","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Asymmetric normally regular digraphs have metric dimension and localization number O(√n log n); certain doubly regular team tournaments improve to O(log n), via probabilistic distinguishing sets.","lead":"The paper proves upper bounds on metric dimension and localization number for families of highly structured digraphs by adapting Babai's probabilistic method for distinguishing sets. These show that highly regular digraphs need only O(√n log n) or even O(log n) probes to locate an invisible robber.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript is a careful, self-contained adaptation of Babai's 1980 probabilistic method to several well-studied families of regular digraphs. The strongest claim (Theorem 2.6) rests on a transparent lower bound for the symmetric difference of in-neighbourhoods that is derived from the defining parameter equation of an ANRD together with a standard hypergraph intersection lemma; both ingredients are correctly invoked and the subsequent application of the distinguishing-set lemma is routine. The same engine produces the sharper O(log n) bound for type-II team tournaments without additional hypotheses. No circularity, hidden parameter restriction, or computational gap appears. The reader's identification of Lemma 2.5 as the load-bearing step is accurate, yet that step survives direct inspection. Therefore the ACCEPT verdict with high confidence remains appropriate; no adjustment is required.","tokens_in":13008,"tokens_out":690,"duration_ms":10874,"concrete_test":"Independently recompute the chain of inequalities in the proof of Lemma 2.5 for the smallest admissible parameters with λ,μ ≥ 1 (e.g., the known ANRD(16,6,2,2) or any DRAD with λ = 1). Verify that |N⁻(u)△N⁻(v)| ≥ √n-1 holds for every pair and that the hypergraph intersection hypothesis of Lemma 2.4 is satisfied; if any pair falls below the bound the O(√n log n) claim fails for that family.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly flags Lemma 2.5 as the key step for the O(√n log n) claim of Theorem 2.6. That lemma's proof is self-contained and holds under the stated hypotheses: the NRD equation (Lemma 2.3) forces k ≥ √n-1 whenever λ,μ ≥ 1; the two elementary inclusions give k-μ ≤ 2(k-λ) and k-λ ≤ 2(k-μ), so |N⁻(u)△N⁻(v)| ≥ k-d with d = min{λ,μ}; Babai's hypergraph lemma then yields k > √(nd) and the elementary inequality √(nd)-d > √n-1 follows from n ≥ 4d. The n ≥ 108 threshold simply ensures √n-1 > 2 log n so that Lemma 2.1 applies. No admissible parameter set is known (or claimed) that would violate these relations while still defining an ANRD, and the paper never asserts the bound outside λ,μ > 0. The same probabilistic engine is applied cleanly to the other families. Consequently the central claims stand.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper adapts Babai’s 1980 probabilistic method for distinguishing sets of undirected strongly regular graphs to several families of highly structured digraphs. After establishing a general existence lemma (Lemma 2.1) that produces a distinguishing set of size O(n log n / c) whenever every pair of vertices has in-neighbourhood symmetric difference at least c, the authors obtain concrete upper bounds on both metric dimension and localization number. The main results are: for an asymmetric normally regular digraph ANRD(n,k,λ,μ) with λ,μ>0 and n≥108 one has ζ(D)≤dim(D)≤⌈2n log n/(√n-1)⌉=O(√n log n) (Theorem 2.6); the same O(log n) bound holds for nearly doubly regular tournaments (Theorem 2.11) and for doubly regular (m,r)-team tournaments of type II (Theorem 2.15). Parallel but weaker bounds are derived for ordinary graphs, Deza digraphs and divisible design digraphs.","tokens_in":13273,"tokens_out":903,"duration_ms":6415,"significance":"The work supplies the first systematic upper bounds on metric dimension and localization number for the principal directed analogues of strongly regular graphs. The O(√n log n) result for ANRDs and the sharper O(log n) results for the two tournament families are natural directed counterparts of Babai’s classical estimates and of the known bounds for Paley graphs and doubly regular tournaments. The proofs are self-contained, rely only on standard parameter equations and a classical hypergraph intersection lemma, and cleanly separate the general probabilistic engine from the family-specific estimates of |N⁻(u)△N⁻(v)|. The paper therefore both extends the undirected theory and provides a reusable template for other digraph families defined by common-neighbour conditions.","major_comments":[],"minor_comments":[{"comment":"In the abstract and the final sentence of the introduction the authors write “common out-neighbours”, yet all subsequent arguments (Lemma 2.1, the definition of distinguishing sets, and every application) work exclusively with in-neighbourhoods N⁻. A single clarifying sentence would remove the inconsistency.","section":null},{"comment":"Theorem 2.2 states the bound in terms of t=min{λ,μ,2λ-μ}, while the paragraph immediately following correctly notes that for ANRDs one may take t=max{λ,μ}. The two statements are compatible but the switch of min/max is easy to miss; a parenthetical remark would help.","section":null},{"comment":"The threshold n≥108 in Theorem 2.6 is only needed to guarantee √n-1>2 log n so that Lemma 2.1 applies. It would be useful to record the (slightly weaker) bound that holds for all n≥4 once the ceiling function is taken into account.","section":null},{"comment":"Figure 1 is a Cayley digraph of Q8; the caption and the surrounding text correctly identify it as an NRD(8,3,1,0), but the figure itself does not label the digons, which may confuse a reader unfamiliar with the quaternion group.","section":null},{"comment":"A few typographical slips: “cope probe” (p. 2), “orG” (p. 2), and the missing space before “and” in the definition of ordinary graphs (p. 7).","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, self-contained contribution that sits comfortably within the scope of a combinatorial journal. The only potential novelty concern is that the O(log n) bound for doubly regular tournaments was already known; the authors correctly present it as a special case of their more general framework, so I see no problem."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The paper does exactly what the abstract promises: it takes Babai's 1980 probabilistic argument for distinguishing sets in strongly regular graphs and rewrites it for digraphs (Lemma 2.1), then verifies the neighbourhood-difference hypothesis for normally regular digraphs, Deza digraphs, divisible design digraphs, nearly doubly regular tournaments and type-II team tournaments. The headline results are Theorem 2.6 (O(√n log n) for asymmetric NRDs with λ, μ > 0) and Theorem 2.15 (O(log n) for type-II team tournaments). Both are new; the earlier special cases for Paley tournaments and DRTs sit inside them as corollaries.\n\nThe proofs are short and self-contained. Once you grant the classical parameter equations (Jørgensen, Crnković–Kharaghani, etc.) and Babai's hypergraph intersection lemma, the lower bounds on |N⁻(u)△N⁻(v)| fall out by elementary counting. Lemma 2.5 is the only non-trivial step for the √n claim; the stress-test note is right that it holds under the stated hypotheses, and the n ≥ 108 threshold is just the point where √n-1 exceeds 2 log n. No circularity, no free parameters, no invented objects.\n\nSoft spots are minor and openly acknowledged by the authors. The method needs a uniform lower bound on the symmetric difference of in-neighbourhoods, so it does not immediately cover digons or directed strongly regular graphs defined via 2-paths rather than common neighbours. The O(√n log n) bound may not be tight; they flag this themselves. Citation pattern is appropriate: Babai, the design-theory sources, and their own earlier work on oriented localization.\n\nThis is solid incremental work for anyone already working on metric dimension or localization games on digraphs. It will not reorganize the field, but it supplies the first general estimates for several natural families and is written cleanly enough that a referee can check every line. I would send it to peer review without hesitation.","headline":"Clean directed adaptation of Babai that delivers the first general O(√n log n) and O(log n) bounds for several design-theoretic digraph families.","tokens_in":13868,"tokens_out":577,"would_cite":true,"duration_ms":4698,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C12","05C20","05C69"],"pacs":[],"model":"grok-4.5","headline":"Highly structured digraphs have small localization numbers and metric dimensions: O(√n log n) for asymmetric normally regular digraphs, and O(log n) for certain team tournaments.","keywords":["metric dimension","localization number","normally regular digraphs","doubly regular tournaments","team tournaments","distinguishing sets","Deza digraphs"],"falsifier":"Exhibit a single asymmetric normally regular digraph on n ≥ 108 vertices with λ, μ > 0 in which two vertices have in-neighbourhoods whose symmetric difference is smaller than √n-1; the O(√n log n) bound then fails for that graph.","tokens_in":13915,"feed_emoji":"→","tokens_out":685,"duration_ms":4818,"temperature":0.7,"pith_summary":"The paper adapts Babai’s 1980 probabilistic method for resolving sets of undirected strongly regular graphs to directed analogues defined by common out-neighbours. It shows that the localization number and metric dimension of an asymmetric normally regular digraph on n vertices are at most O(√n log n), provided every pair of vertices has a positive number of common out-neighbours. The same argument yields O(log n) bounds for nearly doubly regular tournaments and for a class of doubly regular team tournaments. These upper bounds matter because they guarantee that a short list of distance probes can uniquely identify any vertex in these highly regular directed networks, just as in the classical undirected setting.","feed_headline":"Digraphs that look regular need only O(√n log n) probes","feed_subtitle":"Localization number and metric dimension stay small for normally regular digraphs and team tournaments","key_machinery":"A probabilistic existence lemma (Lemma 2.1) that produces a distinguishing set of size O(n log n/c) whenever every pair of vertices has symmetric difference of in-neighbourhoods at least c; the size c is then lower-bounded by √n-1 for ANRDs via the NRD parameter equation and a hypergraph intersection argument.","core_discovery":"For any asymmetric normally regular digraph D on n ≥ 108 vertices with positive parameters λ, μ, the localization number and metric dimension satisfy ζ(D) 孍im(D) 孌eil(2n log n/(√n-1)), hence are O(√n log n). For doubly regular (m,r)-team tournaments of type II the bound improves to O(log n).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Normally regular digraphs need only O(√n log n) for localization","Metric dimension of asymmetric NR digraphs is O(√n log n)","Doubly regular team tournaments resolve in O(log n) probes","Structured digraph families admit small resolving sets via Babai method","Localization number stays O(√n log n) for highly regular digraphs"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that every pair of vertices in an asymmetric normally regular digraph with positive parameters has in-neighbourhoods whose symmetric difference is at least the square root of the number of vertices.","fun_headline_variants_meta":{"raw":{"variants":["Normally regular digraphs need only O(√n log n) for localization","Metric dimension of asymmetric NR digraphs is O(√n log n)","Doubly regular team tournaments resolve in O(log n) probes","Structured digraph families admit small resolving sets via Babai method","Localization number stays O(√n log n) for highly regular digraphs"]},"model":"grok-4.5","effort":"low","cost_usd":0.00527,"raw_usage":{"total_tokens":1355,"prompt_tokens":715,"num_sources_used":0,"completion_tokens":102,"cost_in_usd_ticks":52700000,"prompt_tokens_details":{"text_tokens":715,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":538,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":715,"tokens_out":102,"duration_ms":44075,"temperature":1.0,"reasoning_tokens":538,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T08:00:03.526617+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single asymmetric normally regular digraph on n ≥ 108 vertices with λ, μ > 0 in which two vertices have in-neighbourhoods whose symmetric difference is smaller than √n-1; the O(√n log n) bound then fails for that graph.","supporting_citations":[],"review_version":1}