Partially-Overlapping Minimal Cuts

• Partially-overlapping minimal cuts are minimal separating sets that intersect without one containing the other, impacting graph theory, Feynman integrals, and combinatorial multicuts.
• They are encoded via cactus decompositions that represent cyclic overlap patterns, facilitating efficient enumeration and structural insights in complex networks.
• Their analysis drives algorithmic advances and clarifies singularity structures, influencing multicut enumeration and generalized Steinmann constraints in quantum field theory.

Partially-overlapping minimal cuts are a central concept in the study of graph separation, Feynman integral singularity structure, and combinatorial classification of multicuts. They arise when multiple minimal cut sets share vertices, edges, or propagators in a manner that is neither nested nor disjoint—resulting in intricate interdependencies for both analytic and algorithmic properties. The phenomenon is structurally significant in combinatorics, algebraic geometry, and quantum field theory, as it uniquely impacts enumeration of multicuts, cyclic decompositions of graphs, and generalized Steinmann-type constraints on iterated discontinuities in integral representations.

1. Formal Definitions and Characterization

In the context of graph theory, let $G=(V,E)$ be an undirected, finite or infinite graph. A minimal cut is a set, either of edges or nodes, whose removal disconnects prescribed terminal pairs, and for which no proper subset achieves the same separation. When studying multicuts for general sets of terminal pairs $B \subseteq V \times V$, a cut is partially-overlapping with another if their corresponding sets of edges or nodes have non-trivial intersection and neither is contained in the other, i.e., $S_1 \cap S_2 \neq \emptyset$, $S_1 \setminus S_2 \neq \emptyset$, $S_2 \setminus S_1 \neq \emptyset$ (Kurita et al., 2020).

In Feynman integral analysis, a minimal cut for a singular hypersurface $\lambda(p)=0$ is the smallest set of propagators whose on-shell conditions are necessary and sufficient for the occurrence of the singularity. Two minimal cuts $S_\lambda$, $S_\mu$ are partially overlapping if their intersection is non-empty and neither is strictly contained in the other, an algebraic property that translates to complex restrictions on allowed sequences of singularities (Hannesdottir et al., 2024).

In infinite graphs, particularly those with ends (equivalence classes of rays), two minimal edge-cuts $K,L$ are partially overlapping ("crossing") when the four intersections between their associated partitions of ends are all non-empty. This crossing ensures a non-hierarchical relation among the cuts and organizes them into cyclic sets (Evangelidou et al., 2011).

2. Structural and Cactus Decomposition

The theory of cactus representations provides a profound encoding mechanism for families of minimal edge-cuts with arbitrary overlaps. A cactus is a graph where cycles intersect in at most one vertex, and the minimal edge-cuts of the original graph map bijectively to those of the cactus. The A cactus theorem for end cuts demonstrates that all minimal cuts (including partially-overlapping ones) can be grouped into either isolated cuts or maximal cyclic sets. Each cyclic set corresponds to a cycle in the cactus, and crossing (partially-overlapping) cuts necessarily generate such cycles (Evangelidou et al., 2011).

Key lemmas establish that any two crossing cuts (partially-overlapping) of cardinality $n=2k$ are disjoint, each meets the other's components in exactly $k$ edges, and their union partitions the graph into four non-empty components, each containing an end. These sets are extended, by closure under crossing, into unique maximal cyclic sets, which form the combinatorial foundation for the structure of the cactus.

3. Enumeration and Complexity in Multicuts

Algorithmic frameworks address the enumeration of minimal multicuts where overlapping demands are present. The incremental polynomial-delay approach for node multicuts constructs a supergraph of solutions where each vertex is a minimal cut, and edges represent local modifications. The traversal algorithms guarantee that partially-overlapping cuts are correctly enumerated by employing local reductions, contraction, and separator enumeration on induced subgraphs. The essential characterization ensures that each cut corresponds uniquely to a partition of terminal-containing components, and overlaps are managed through subproblems on star-demand structures (Kurita et al., 2020).

Complexity remains controlled (polynomial delay), even for instances of severe overlap, by reduction strategies that detect forced vertices and contract adjacently connected terminals, reducing the solution space without compromising completeness or minimality. In multiway cuts (where B is complete), proximity and reverse search paradigms permit efficient enumeration with exponential or polynomial space guarantees, again adapting to overlapping situations by maintaining distance or depth measures in the solution graph.

4. Hierarchical and Genealogical Constraints in Feynman Integrals

Partially-overlapping minimal cuts exert pronounced effects on the analytic structure of Feynman integrals. The genealogical constraints framework generalizes Steinmann relations, showing that for any pair of partially-overlapping cuts $(S_\lambda, S_\mu)$, the double discontinuity $\Delta_\mu\Delta_\lambda I$ vanishes at all orders—regardless of adjacency or separation in the sequence. This arises because after pinching the contours for $S_\lambda$, the corresponding directions in Feynman-parameter space are inaccessible for subsequent pinching required by $S_\mu$. The Euler-characteristic drop argument in the blown-up Landau equations domain allows the derivation of these constraints without solving the full Landau system (Hannesdottir et al., 2024).

Genealogical constraints rigorously rule out sequences of singularities that would otherwise be plausible and, at the level of symbol alphabets in polylogarithmic integrals, dramatically reduce the allowed spaces, as illustrated in multiloop examples. This tightening surpasses extended Steinmann relations and holds universally across loop orders, particle multiplicities, and mass configurations.

5. Steinmann Relations and Violations

The interplay between minimal cut overlap and Steinmann relations is explicitly mapped in Feynman integral theory. For massless two-loop integrals, the existence of partially-overlapping minimal cuts (particularly those combining to form specified "box-like" four-propagator subgraphs) constitutes the only known mechanism for Steinmann violation. A graphical test, checking for such subgraphs in the union of two minimal cuts, efficiently predicts all observed Steinmann violations in two-loop five-particle scattering (Hannesdottir et al., 30 Dec 2025).

Concretely, in topologies like the one-mass pentabox or two-mass acnode, partially-overlapping cuts yield double discontinuities in supposedly forbidden channel pairs. Conversely, absence of such subgraphs ensures preservation of Steinmann relations. This suggests that partially-overlapping minimal cuts not only underlie analytic pathologies but also provide a complete classification tool for their occurrence.

6. Applications and Impact Across Domains

The study and algorithmic management of partially-overlapping minimal cuts have broad applications. In combinatorics and algorithmic graph theory, they underlie enumeration, optimization, and structural decomposition tasks in network reliability, multicriteria flow, and end-separation. In mathematical physics, they control singular regions in Feynman integrals, dictate symbol alphabets in polylogarithmic representations, and define all-orders constraints on discontinuity sequences that impact amplitude bootstrap techniques.

Their encoding via cactus structures, efficient enumeration in the presence of complex overlaps, and role in analytic singularity hierarchies collectively illuminate deep interconnections among combinatorial graph theory, algebraic analysis, and quantum field theory.