Bidirected Graphs

• Bidirected graphs are defined as combinatorial structures that attach independent '+' or '−' orientations to each end of an edge.
• They generalize both directed and signed graphs, enabling advanced modeling of connectivity, flows, and network decompositions.
• They underpin diverse applications in combinatorial optimization, statistical inference, and genome graph analysis through algorithmic transformations.

A bidirected graph is a combinatorial structure generalizing both directed and signed graphs by attaching an independent orientation (“+” or “−”) to each end of each edge. This capability to locally encode “incoming” and “outgoing” behavior at each vertex-end makes bidirected graphs a versatile framework in combinatorial optimization, algebra, network theory, and modern applications such as biological sequence graphs and statistical models.

1. Formal Definition and Basic Properties

Let $G=(V,E)$ be a finite undirected graph (possibly with loops or parallel edges). A bidirected structure on $G$ comprises, for each edge $e$ with ends $u$ and $v$, two sign assignments $\epsilon, \epsilon' \in \{+1, -1\}$, not necessarily of opposite sign. These are sometimes encoded through functions:

• $\beta(e,u), \beta(e,v) \in \{+,-\}$ assigning a sign to each end of $e$ (Sampathkumar et al., 2013, Bessouf et al., 2016, Kita, 2017, González et al., 2023).
• Alternatively, a bidirection (in the sense of Edmonds) $\tau:\{(e,x): x \in V,\, e \ni x\} \to \{+1, -1\}$ (Gellert et al., 2015, Bessouf et al., 2016).

Each edge is then described as the unordered pair $\{\,(u,\alpha),(v,\beta)\,\}$ or simply as $G$0, where $G$1. Notable specializations:

• $G$2 or $G$3 at an edge correspond to a classic directed arc from head to tail or vice versa.
• $G$4 and $G$5 encode parallel signed/arbitrarily oriented edges, as in signed graphs.

A walk in a bidirected graph is a sequence of signed vertices $G$6, such that for each consecutive pair, $G$7 is an edge with $G$8 and $G$9 (Schmidt et al., 13 May 2026). At each internal vertex, incident half-edges must carry opposite signs.

Bidirected graphs are thus a common generalization of digraphs (all edges have ends of opposite sign) and signed graphs (ends at both vertices share the same sign) (Kita, 2017, Sampathkumar et al., 2013, González et al., 2023).

2. Connections to Directed and Signed Graphs

Bidirected graphs unify and extend orientations from digraphs and edge signings from signed graphs:

• A digraph corresponds to the case where every edge has one $e$0 and one $e$1 sign (local orientation). The walk structure thus coincides with classical directed walks (Kita, 2017, 1310.00179, Ghorbani et al., 15 Nov 2025).
• A signed graph, as per Zaslavsky, arises as the image of a bidirected graph where both ends of every edge share the same sign. The signed graph induced by a bidirected structure uses the rule $e$2 (Sampathkumar et al., 2013).
• "Directionally two-signed" graphs are equivalent to bidirected graphs under a mapping between ordered pairs of endpoint-signs and bidirection assignments (Sampathkumar et al., 2013).

This generality allows representation of a richer class of combinatorial objects and supports structure-preserving mappings between signatures, orientations, and degree patterns (Gellert et al., 2015, González et al., 2023).

3. Path Structures, Strong Connectivity, and Decomposition

Standard walk, trail, and cycle notions generalize through local sign alternation. A ditrail is a trail in which the edges at each inner vertex have opposite incident signs. A dipath is a ditrail with no repeated vertices (Kita, 2017).

Unlike digraphs, reachability by trails and by paths may diverge for bidirected graphs; ditrails can self-intersect, causing trail-path divergence (Kita, 2017).

The strong connectivity theory is more intricate:

• Bidirected strong components are defined using circular connectivity: two vertices are circularly connected (denoted $e$3) if a path between them exists using only edges that belong to cyclic ditrails (Kita, 2017).
• The general Kotzig–Lovász decomposition partitions circular components into equivalence classes via a relation $e$4, based on the absence of closed $e$5-ditrails (Kita, 2017). This unifies strong component partitioning in digraphs with matching/block decompositions in factor theory.

Radials and semiradials offer further analogues of critical graphs and flowgraphs, giving constructive decompositions via iterative addition of "diears" (directed ears) and controlling access to the root by specific sign-sequence restrictions (Kita, 2019).

4. Structural Theorems, Flows, and Connectivity

Bidirected graphs exhibit fundamental differences in the context of classical connectivity theorems and flows:

• The classical Menger theorem does not always hold verbatim. It requires a generalization to account for "turnarounds": unions of $e$6-$e$7 and $e$8-$e$9 paths (max-packing/min-cuts via turnarounds, rather than only $u$0-$u$1 paths) (Ghorbani et al., 15 Nov 2025).
• Rooted connectivity and edge/path-disjointness in bidirected graphs admit a reduction, under suitable “cleanliness” hypotheses, to the corresponding results in digraphs via a canonical edge decomposition and contraction (Abrishami et al., 27 Sep 2025).

Flow theory on bidirected graphs generalizes group-valued flows and supports dualities with tensions and colourings via local sign combinations and orientations (DeVos, 2013, DeVos et al., 2023). Bouchet’s conjecture concerns the existence of nowhere-zero 6-flows in every flow-admissible bidirected graph. Improvements (e.g., 8-flow for cyclically 5-edge-connected graphs, 12-flow in general) are achieved using modular lifting arguments, parity decompositions, and perfect matching theory (DeVos, 2013, DeVos et al., 2023).

Network flows, degree-constrained flows, and factor problems translate bidirected inputs directly to signed incidence matrices and corresponding quadratic forms (incidence forms), allowing algebraic classification, universality results, and root system analysis (González et al., 2023).

5. Algorithmic Aspects and Transformations

Bidirected graphs admit several constructive reductions and algorithmic frameworks:

• Graph Doubling: A canonical transformation converts a bidirected graph $u$2 to a directed graph $u$3 on doubled signed vertices, preserving reachability. This enables the porting of directed-graph algorithms, including shortest paths, flows, and structural enumeration (e.g., ultrabubbles in genomics) to the bidirected field (Schmidt et al., 13 May 2026).
• Transitive closure and reduction: Generalizing Aho–Garey–Ullman for digraphs, bipaths and bicircuits are used to define closure and irredundant spanning subgraphs, respecting signed-matroidal dependencies (Bessouf et al., 2016).
• Degree sequence characterizations: Erdős-Gallai–type criteria, Havel-Hakimi–type moves, and extremal/minimum instances extend to bidirected settings, and the set of all net-degree vectors forms a zonotope with simple bounding facet descriptions (Gellert et al., 2015).

Tables of bidirected graph classes and key results:

Bidirected Specialization	Corresponds to	Key Results/Sources
Digraph (one $u$4 per edge)	Directed graph	Standard reachability
Signed (same sign both ends)	Signed graph	Induced signed graph theory
Generic (all 4 pairs allowed)	Bidirected graph	Trail/path divergence, generalized flows, connectivity decompositions

6. Applications in Statistics, Combinatorics, and Algebra

• Graphical models/statistical inference: Bidirected graphs model systems where absence of bidirected edges encodes marginal independence (covariance structure) (Malouche et al., 2013). Duality with undirected models enables systematic transfer of Markov properties, separation criteria, and faithfulness results. Gaussian linear systems and algebraic walk-matrix frameworks extend natural interpretations, e.g., latent projections corresponding to confounding (Zhao, 2024).
• Incidence quadratic forms and root systems: Every bidirected graph defines an incidence form $u$5, whose structure encodes root systems of Dynkin type $u$6, $u$7, $u$8 depending on the presence of bidirected loops or cycles. These forms classify possible algebraic and combinatorial symmetries of a graph, with applications to gentle algebras and Diophantine equations (González et al., 2023).
• Genome graph analysis: Ultrabubbles in bidirected graphs correspond to biologically relevant loci, and their enumeration via graph doubling links deep combinatorial constraints with practical algorithmic frameworks (Schmidt et al., 13 May 2026).

7. Future Directions and Open Problems

Several prominent research directions are emerging:

• Extensions of graph doubling to more complex signed or higher-dimensional structures (snarls, cacti-decompositions) and to dynamic/streaming graphs (Schmidt et al., 13 May 2026).
• Combinatorial and algorithmic refinements for network optimization, such as minimizing structural augmentation for strong connectivity or optimal edge-modifications (Matsuoka et al., 2017).
• Full characterization of edge-disjoint analogues of classical connectivity theorems, weighted versions, and fractional packings/cuts in the bidirected setting (Ghorbani et al., 15 Nov 2025).
• Systematic exploitation of the duality between undirected and bidirected models in graphical inference, especially for faithfulness and causal identification (Malouche et al., 2013, Zhao, 2024).

A plausible implication is that bidirected graphs will continue to serve as a key combinatorial abstraction for problems requiring local orientation and flexible modeling of flows, connectivity, and statistical dependence, especially in contexts where neither directed nor undirected graphs suffice.