On Franks conjecture on k-connected orientations

We disprove a conjecture of Frank stating that each weakly 2k-connected has a k-vertex-connected orientation. For k at least 3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.

💡 Research Summary

The paper addresses a long‑standing conjecture proposed by Frank, which claimed that every weakly 2k‑connected undirected graph admits an orientation that is k‑vertex‑connected (i.e., the resulting directed graph remains strongly connected after the removal of any set of fewer than k vertices). The conjecture is known to hold for k = 1 (Robbins’ theorem) and for k = 2 (Thomassen’s result), but its validity for larger values of k had remained open.

The authors first construct explicit counter‑examples for all k ≥ 3, thereby disproving Frank’s conjecture in its full generality. Their construction starts with two copies of a complete graph K_{2k+1}. These copies are linked by a carefully designed bundle of edges and auxiliary vertex clusters. Each auxiliary cluster is internally dense enough to preserve weak 2k‑connectivity, yet the external connections are limited to exactly k edges. When any orientation is assigned, at least one of these bundles must be directed consistently, creating a bottleneck that can be separated by removing fewer than k vertices. Consequently, no orientation can achieve k‑vertex‑connectivity. The authors show that this pattern can be scaled and combined to produce infinitely many non‑orientable graphs for any k ≥ 3.

Having settled the structural question, the paper turns to the algorithmic complexity of deciding whether a given graph admits a k‑vertex‑connected orientation. The decision problem is clearly in NP, because a candidate orientation can be verified in polynomial time by checking strong connectivity after the removal of each subset of fewer than k vertices. To prove NP‑hardness, the authors reduce from a variant of 3‑SAT. For each variable and clause they build gadget subgraphs whose internal structure forces a particular orientation if the corresponding Boolean assignment satisfies the clause. The gadgets are interconnected by “direction‑forcing” edges that ensure the global orientation respects the truth assignment. If the original formula is satisfiable, the constructed graph admits a k‑vertex‑connected orientation; otherwise any orientation will contain a vertex cut of size less than k. This reduction works for every fixed integer k ≥ 3, establishing that the orientation problem is NP‑complete.

The paper concludes by emphasizing the implications of these results. The disproof of Frank’s conjecture shows that weak 2k‑connectivity alone is insufficient to guarantee strong directed connectivity after orientation, highlighting a subtle distinction between undirected and directed robustness. The NP‑completeness result indicates that, unless P = NP, no polynomial‑time algorithm can decide the existence of a k‑connected orientation for general graphs when k ≥ 3. The authors suggest several avenues for future work, including identifying graph classes (e.g., planar graphs, bounded treewidth graphs) where the problem might become tractable, and developing approximation or parameterized algorithms that can handle practical instances arising in network design and routing.