Connectivity Under Weakened Bridge Addability

Definitions and mathematical statement

In the conventional setting of finite simple undirected graphs, a class $\mathcal{G}$ is bridge-addable if, for every $G\in\mathcal{G}$ and every pair of vertices $u,v\in V(G)$ in distinct connected components, the graph $G+uv$ obtained by adding the edge $uv$ also belongs to $\mathcal{G}$. The added edge is a bridge: deleting it increases the number of connected components. The supplied record does not specify whether this conventional graph setting, or a labelled or unlabelled version of it, is intended; these conventions require confirmation.

A condition $\mathsf{C}$ on graph classes is weaker than bridge-addability if every bridge-addable class satisfies $\mathsf{C}$; it is strictly weaker if the converse fails.

Source question. Which connectivity bounds and related asymptotic properties established for bridge-addable graph classes remain valid under specified weaker closure conditions?

This is a research question requiring further specification, rather than a single fully quantified assertion. The source does not identify the weaker closure condition, the connectivity bounds to be retained, or the related asymptotic properties. If connectivity refers to the probability that a random graph from a class is connected, the underlying probability distribution, admissible graph orders, nonemptiness assumptions, and limiting regime must also be specified. No particular distribution, bound, or closure condition is imposed here. The supplied background mentions relatively bridge-addable classes as a direction with partial results, but gives no definition or precise result for that notion.

Short English statement

The question asks which connectivity bounds and other asymptotic conclusions survive when closure under adding edges between connected components is weakened. The supplied record leaves both the replacement condition and the target conclusions unspecified, so a precise theorem statement requires further source clarification.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30002543$, title Connectivity Under Weakened Bridge Addability, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied report attribution: Enumerative Combinatorics (2014), Oberwolfach Reports, pp. 635--720, DOI 10.4171/owr/2014/12. https://doi.org/10.4171/owr/2014/12. The background describes this collection as extracted from workshop open-problem discussions. No individual proposer is supplied.
• Supplied literature source: McDiarmid--Weller (2017). https://doi.org/10.1017/S0963548317000128.
• Supplied literature source: Weller thesis. https://ora.ox.ac.uk/objects/uuid%3A667a139e-6d2c-4f67-8487-04c3a0136226.
• Supplied literature source: Bridge-addable connectivity. https://arxiv.org/abs/1504.06344.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open, reports substantial partial results for weaker notions such as relatively bridge-addable classes, and reports no general replacement theorem or classification answering the extracted broad question. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that notation, typography, and wording were normalized without changing mathematical content. These are supplied editorial claims, not independent verification of the formulation in this draft.