Fully Dynamic Breadth First Search and Spanning Trees in Directed Graphs

Gregory Morse; Tam\'as Kozsik

arxiv: 2604.12370 · v1 · submitted 2026-04-14 · 💻 cs.DS

Fully Dynamic Breadth First Search and Spanning Trees in Directed Graphs

Gregory Morse , Tam\'as Kozsik This is my paper

Pith reviewed 2026-05-10 14:40 UTC · model grok-4.3

classification 💻 cs.DS

keywords directeddynamicfullymaintainingspanningtreegraphssingle-source

0 comments

The pith

A framework for fully dynamic BFS trees, levels, and reachability in directed graphs under edge insertions and deletions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Graphs with directions model one-way connections such as web links or traffic rules. Breadth-first search finds the shortest paths from a starting point by exploring level by level. When the graph changes, restarting the search from scratch is slow. The work aims to update the tree structure, the distance levels, and the ordering of nodes efficiently after each change, handling both additions and removals.

Core claim

This preprint presents a framework for fully dynamic BFS in directed graphs together with supporting data structures for maintaining the BFS tree, single-source shortest paths, and single-source reachability under both insertions and deletions.

Load-bearing premise

That the proposed data structures can maintain the BFS tree, level information, and numbering together under arbitrary sequences of insertions and deletions without degrading to static recomputation.

read the original abstract

We study the problem of maintaining a breadth-first spanning tree and the induced BFS ordering in a directed graph under edge updates. While semi-dynamic algorithms are known, maintaining the spanning tree, level information, and numbering together in the fully dynamic setting is less developed. This preprint presents a framework for fully dynamic BFS in directed graphs together with supporting data structures for maintaining the BFS tree, single-source shortest paths, and single-source reachability under both insertions and deletions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This preprint claims a framework for fully dynamic BFS tree maintenance in directed graphs but supplies no algorithms, proofs, or update procedures, leaving the central claims impossible to check.

read the letter

Hi, the one thing to know is that the paper positions itself as the first to maintain a BFS spanning tree, levels, and ordering together under arbitrary insertions and deletions in directed graphs. If the supporting structures actually work without falling back to static recomputation, it would fill a documented gap left by existing semi-dynamic results. Right now that remains a claim rather than a demonstrated result. What is new is the explicit combination of tree maintenance, level information, and numbering in the fully dynamic directed setting, along with the link to single-source shortest paths and reachability. The abstract does a clean job of naming the prior semi-dynamic work and stating why the fully dynamic version is less developed. The soft spots are substantial and central. No data structures, no update rules, no time bounds, and no invariant arguments appear. The weakest assumption is that level changes and renumberings can be propagated correctly across directed reachability shifts without local inconsistencies or global restarts. Deletions in directed graphs can disconnect paths in ways that require careful handling of multiple incoming edges and ordering preservation; nothing in the available text shows how this is solved. Because only the abstract was accessible, I cannot tell whether the framework avoids these pitfalls or simply restates the problem. This is aimed at people working on dynamic graph primitives who need reliable reachability or distance maintenance in changing directed networks. A reader already following that literature would want the full version to see whether the data structures deliver the promised bounds. It deserves a serious referee once the complete manuscript with algorithms and proofs is submitted, because the topic is relevant and the gap it targets is real. Without those details, though, it is not yet ready for review.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to study the maintenance of a breadth-first spanning tree and induced BFS ordering in directed graphs under edge updates. It notes that semi-dynamic algorithms exist but fully dynamic maintenance of the spanning tree, level information, and numbering is less developed, and presents a framework for fully dynamic BFS together with supporting data structures for the BFS tree, single-source shortest paths, and single-source reachability under insertions and deletions.

Significance. If the claimed framework and data structures correctly maintain the required properties under arbitrary updates without degrading to static recomputation, the result would address an important gap in dynamic graph algorithms, as fully dynamic BFS and reachability in directed graphs have seen limited progress compared to semi-dynamic settings. This could enable more efficient handling of dynamic networks for shortest paths and connectivity queries.

major comments (1)

[Abstract] The manuscript provides only a high-level abstract with no algorithmic descriptions, pseudocode, invariants, update procedures, time bounds, or proofs. Without these, it is impossible to assess whether the supporting data structures correctly propagate level changes and renumberings under insertions and deletions, as required for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and constructive feedback on our manuscript. We address the major comment below and commit to revisions that strengthen the presentation.

read point-by-point responses

Referee: [Abstract] The manuscript provides only a high-level abstract with no algorithmic descriptions, pseudocode, invariants, update procedures, time bounds, or proofs. Without these, it is impossible to assess whether the supporting data structures correctly propagate level changes and renumberings under insertions and deletions, as required for the central claim.

Authors: We acknowledge that the current abstract is high-level and that the manuscript as presented does not include pseudocode, explicit invariants, detailed update procedures, time bounds, or proofs. The full text describes a framework for maintaining BFS trees, levels, and reachability under insertions and deletions in directed graphs, but to enable proper assessment of correctness (including propagation of level changes and renumberings), we will expand the revised version with these elements: pseudocode for the core update routines, stated invariants for the data structures, explicit time bounds, and proof outlines showing how the structures handle arbitrary edge updates without falling back to static recomputation. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract and available context present an algorithmic framework for fully dynamic BFS, spanning trees, and reachability maintenance under insertions/deletions in directed graphs. No equations, parameters, derivations, self-citations, or ansatzes are described that could reduce a claimed result to its inputs by construction. The contribution is a data-structure design rather than a mathematical prediction or uniqueness theorem, so no load-bearing steps exhibit self-definitional, fitted-input, or citation-chain circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5360 in / 861 out tokens · 23328 ms · 2026-05-10T14:40:50.902354+00:00 · methodology

Fully Dynamic Breadth First Search and Spanning Trees in Directed Graphs

Core claim

Load-bearing premise

discussion (0)