Characterization of Vulnerable st-Graphs

Characterize all vulnerable st-graphs, namely, all two-terminal directed multigraphs that admit latency-function assignments producing the Braess paradox, beyond the undirected and irredundant directed cases currently characterized.

Background

Vulnerability asks whether a network topology admits some assignment of edge latency functions and a traffic demand for which removing edges decreases the Wardrop-equilibrium latency. The paper explains that Roughgarden introduced this graph-theoretic perspective and left the general characterization of vulnerable st-graphs unresolved.

Subsequent work cited in the paper characterizes vulnerable undirected multigraphs and irredundant directed multigraphs, while general directed graphs require consideration of their maximum irredundant subgraphs. The stated problem concerns the broader characterization problem identified in the cited earlier work.

References

A change of perspective suggested in is to study the Braess paradox from a graph-theoretical point of view: in particular, a problem left open in is the characterization of {\em vulnerable} $st$-graphs, which are those that admit instances (i.e., assignments of latency functions to edges) that generate the paradox.

— An Incremental Algorithm for Checking the Possibility of Braess Paradox in Dynamic Nets  (2609.24719 - Fiorenza et al., 21 Sep 2026) in Section 1, paragraph beginning “Braess paradox has been studied for decades”

In contrast, the fully dynamic case (i.e., when edge additions and removals can arbitrarily interleave) seems to be a more complex problem, since it is not clear what information should be kept from one addition/deletion to the next one to enable an efficient vulnerability checking.

— An Incremental Algorithm for Checking the Possibility of Braess Paradox in Dynamic Nets  (2609.24719 - Fiorenza et al., 21 Sep 2026) in Section 5, Conclusions