Skew-symmetric graph¶

In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12073
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Graph Theory → Mathematics

Core Idea¶

Skew-symmetric graph is treated here as the recurring computer science and information systems identity summarized by this source-grounded definition: In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.

In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs. Skew-symmetric graphs were first introduced under the name of antisymmetrical digraphs by , later as the double covering graphs of polar graphs by , and still later as the double covering graphs of bidirected graphs by.

They arise in modeling the search for alternating paths and alternating cycles in algorithms for finding matchings in graphs, in testing whether a still life pattern in Conway's Game of Life may be partitioned into simpler components, in graph drawing, and in the implication graphs used to efficiently solve the 2-satisfiability problem. A skew-symmetric graph may equivalently be defined as the double covering graph of a polar graph or switch graph, which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation-reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs.

For Skew-symmetric graph, the abstraction is narrower than the article's general subject matter: a positive case must preserve In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in computer science and information systems, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The transpose graph of G is the graph formed by reversing every edge of G, and σ defines a graph isomorphism from G to its transpose.
Constitutive relation — If such a partition exists, a satisfying assignment may be formed by assigning a true value to every variable in S and a false value to every variable in σ(S).
Operating condition — As he shows, for switch graphs with at most three edges per vertex, this may be tested in polynomial time by repeatedly removing bridges (edges the removal of which disconnects the graph) and vertices at which all edges belong to a single partition until no more such simplifications may be performed.
Recognition evidence — An instance of the 2-satisfiability problem, that is, a Boolean expression in conjunctive normal form with two variables or negations of variables per clause, may be transformed into an implication graph by replacing each clause \scriptstyle u\lor v by the two implications.
Admissible variation — In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.
Characteristic consequence — As defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties.
Failure boundary — This equivalence is the one used by to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges.

What It Is Not¶

Not the whole field of computer science and information systems. The node requires the specific identity stated by In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.
Not an over-broad reading. However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism.
Not an over-broad reading. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation-reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs.
Not an over-broad reading. This complexity does not affect path-finding algorithms for skew-symmetric graphs, because these algorithms assume that the skew-symmetric structure is given as part of the input to the algorithm rather than requiring it to be inferred from the graph alone.
Not automatically Semi-symmetric graph. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Skew-symmetric graph applies literally inside computer science and information systems wherever the source-defined carrier and relation can be established. Its documented habitats include:

Polar/switch graphs, double covering graphs, and bidire. This equivalence is the one used by to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges.
Definition. As defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties.
Definition. One may use the third property to extend σ to an orientation-reversing function on the edges of G.
Matching. If the length function is allowed to have negative lengths, the existence of a negative regular cycle may be tested in polynomial time.
Matching. Given additionally a non-negative length function on the edges of the graph that assigns the same length to any edge e and to σ(e), the shortest regular path connecting a given pair of nodes in a skew-symmetric graph with m edges and n vertices may be tested in time O(m log n).
Documented setting. They arise in modeling the search for alternating paths and alternating cycles in algorithms for finding matchings in graphs, in testing whether a still life pattern in Conway's Game of Life may be partitioned into simpler components, in graph drawing, and in the implication graphs used to efficiently solve the 2-satisfiability problem.

Outside computer science and information systems, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Skew-symmetric graph names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. The strongest recognition evidence in the frozen account is: An instance of the 2-satisfiability problem, that is, a Boolean expression in conjunctive normal form with two variables or negations of variables per clause, may be transformed into an implication graph by replacing each clause \scriptstyle u\lor v by the two implications. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Skew-symmetric graph compresses multiple computer science and information systems details into a stable diagnostic relation. The source shows both the central mechanism—if such a partition exists, a satisfying assignment may be formed by assigning a true value to every variable in S and a false value to every variable in σ(S).—and the practical consequence—as defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the computer science and information systems entities to which the claim applies.
State the relation. Use the source-grounded identity: In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.
Check operation and conditions. As he shows, for switch graphs with at most three edges per vertex, this may be tested in polynomial time by repeatedly removing bridges (edges the removal of which disconnects the graph) and vertices at which all edges belong to a single partition until no more such simplifications may be performed.
Demand recognition evidence. An instance of the 2-satisfiability problem, that is, a Boolean expression in conjunctive normal form with two variables or negations of variables per clause, may be transformed into an implication graph by replacing each clause \scriptstyle u\lor v by the two implications.
Test variation. Change an implementation or setting while preserving in graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Skew-symmetric graph transfers literally when a new case preserves the same carrier type, relation, and recognition test. This equivalence is the one used by to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges. As defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties.

Beyond the home domain. No canonical parent is asserted for Skew-symmetric graph. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

As defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points; recognition evidence → An instance of the 2-satisfiability problem, that is, a Boolean expression in conjunctive normal form with two variables or negations of variables per clause, may be transformed into an implication graph by replacing each clause \scriptstyle u\lor v by the two implications

Applied / In Practice¶

In this case, the number of different mappings σ that realize the skew symmetry of the graph equals half the length of the cycle. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Examples; invariant → In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points; boundary → the case exits the class when however, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism

Structural Tensions¶

T1 — Stable identity versus admissible variation. However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation-reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. This complexity does not affect path-finding algorithms for skew-symmetric graphs, because these algorithms assume that the skew-symmetric structure is given as part of the input to the algorithm rather than requiring it to be inferred from the graph alone. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. A bidirected graph may be interpreted as a polar graph by letting the partition of edges at each vertex be determined by the partition of endpoints at that vertex into heads and tails; however, swapping the roles of heads and tails at a single vertex ("switching" the vertex) produces a different bidirected graph but the same polar graph. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The transpose graph of G is the graph formed by reversing every edge of G, and σ defines a graph isomorphism from G to its transpose. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Skew-symmetric graph literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. If such a partition exists, a satisfying assignment may be formed by assigning a true value to every variable in S and a false value to every variable in σ(S). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Skew-symmetric graph distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Skew-symmetric graph is structural-leaning. Its structural side is the repeatable organization summarized by In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Its framed side is the computer science and information systems vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: As he shows, for switch graphs with at most three edges per vertex, this may be tested in polynomial time by repeatedly removing bridges (edges the removal of which disconnects the graph) and vertices at which all edges belong to a single partition until no more such simplifications may be performed. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The transpose graph of G is the graph formed by reversing every edge of G, and σ defines a graph isomorphism from G to its transpose. If such a partition exists, a satisfying assignment may be formed by assigning a true value to every variable in S and a false value to every variable in σ(S). It further constrains recognition and variation through: As he shows, for switch graphs with at most three edges per vertex, this may be tested in polynomial time by repeatedly removing bridges (edges the removal of which disconnects the graph) and vertices at which all edges belong to a single partition until no more such simplifications may be performed. An instance of the 2-satisfiability problem, that is, a Boolean expression in conjunctive normal form with two variables or negations of variables per clause, may be transformed into an implication graph by replacing each clause \scriptstyle u\lor v by the two implications.

What is domain-bound. computer science and information systems supplies the operative entities, technical vocabulary, warrants, and exceptions that make Skew-symmetric graph literal. Its documented scope includes the condition that This equivalence is the one used by to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges. Another bounded application condition is that As defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Network.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Skew-symmetric graph. The reviewed identity is: In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Skew-symmetric graph Domain-specific

Parents (1) — more general patterns this builds on

Skew-symmetric graph is a kind of Network Prime

Skew-symmetric graph is a domain-specific kind of network under its frozen identity and differentia.

Hierarchy path (1) — routes to 1 parentless root

Skew-symmetric graph → Network → Reservoir-Flux Network → Conservation Laws → Invariance

Neighborhood in Abstraction Space¶

Skew-symmetric graph sits in a moderately populated region (41^st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Data Structures & Graph Variants (17 abstractions)

Nearest neighbors

Block Graph — 0.88
Graph Toughness — 0.88
Split graph — 0.88
Maximal independent set — 0.87
Graph isomorphism — 0.86

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points?
Semi-symmetric graph. A regular undirected graph whose automorphism group is transitive on edges but not on vertices. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Homogeneous Graph. A graph whose every isomorphism between finite induced subgraphs extends to an automorphism of the entire graph, making all finite local copies globally interchangeable. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Zero-symmetric graph. A connected cubic graph whose automorphism group acts regularly on vertices but is not transitive on edges. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Skew-symmetric graph remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside computer science and information systems lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Skew-symmetric_graph (revision 1363391916).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.