Strong CR Tournaments in Determinant Theory

• Strong CR tournaments are tournaments defined using determinant layers where each subtournament’s determinant is strictly bounded, ensuring rigidity via switching and transitive blowups.
• They are characterized by CR-associated vertices and extension rules, with canonical templates like the Lₙ family serving as irreducible building blocks in the determinant layer Dₖ.
• Their structure implies that any tournament containing a basic strong CR tournament is switching equivalent to a transitive blowup, preserving the determinant-bound property.

Strong CR tournaments are a class of tournaments introduced in the determinant-based study of skew-adjacency matrices. In this setting, if $T$ is a tournament with skew-adjacency matrix $S_T$, the determinant of $T$ is $\det(T):=\det(S_T)$, and for a positive odd integer $k$ one considers the class $\mathcal D_k$ of tournaments all of whose subtournaments have determinant at most $k^2$. A CR tournament is a tournament in $\mathcal D_k\setminus \mathcal D_{k-2}$ whose one-vertex extensions preserve the determinant bound only in the “CR-associated” directions; a strong CR tournament is a CR tournament for which every $1$-transitive blowup is again a CR tournament. The central role of strong CR tournaments is that, for a basic strong CR tournament $H$, every tournament $S_T$0 containing a subtournament switching isomorphic to $S_T$1 remains in the same determinant layer $S_T$2 if and only if $S_T$3 is switching equivalent to a transitive blowup of $S_T$4 (Zeng et al., 10 Aug 2025).

1. Determinant framework and switching structure

A tournament is an orientation of the complete graph: for every two distinct vertices $S_T$5, exactly one of $S_T$6 or $S_T$7 holds. If $S_T$8 is an $S_T$9-tournament with vertex ordering $T$0, its adjacency matrix is

$T$1

and its skew-adjacency matrix is

$T$2

The determinant of a tournament is

$T$3

This does not depend on the chosen vertex ordering. The determinant is $T$4 for tournaments of odd order, while for even order it is the square of an odd integer (Zeng et al., 10 Aug 2025).

For a positive odd integer $T$5,

$T$6

The layer

$T$7

consists of tournaments whose subtournament determinants are all $T$8, and for which at least one subtournament has determinant exactly $T$9. The convention

$\det(T):=\det(S_T)$0

is also used.

Switching is fundamental. Given $\det(T):=\det(S_T)$1, the switch of $\det(T):=\det(S_T)$2 with respect to $\det(T):=\det(S_T)$3 is obtained by reversing all arcs between $\det(T):=\det(S_T)$4 and $\det(T):=\det(S_T)$5. Two tournaments are switching equivalent if one is obtained from the other by such a switch. Determinant and membership in $\det(T):=\det(S_T)$6 are invariant under switching. A tournament $\det(T):=\det(S_T)$7 is switching isomorphic to $\det(T):=\det(S_T)$8 if some switch of $\det(T):=\det(S_T)$9 is isomorphic to $k$0.

Blowups provide the main constructive operation. If $k$1 has vertices $k$2, and $k$3 are tournaments, the blowup

$k$4

is obtained by replacing $k$5 by $k$6, and orienting all edges between $k$7 and $k$8 according to the arc $k$9 in $\mathcal D_k$0. If each $\mathcal D_k$1 is transitive and $\mathcal D_k$2, this is the transitive $\mathcal D_k$3-blowup of $\mathcal D_k$4, denoted

$\mathcal D_k$5

A $\mathcal D_k$6-transitive blowup means exactly one $\mathcal D_k$7 and all others are $\mathcal D_k$8. A key inherited fact is that

$\mathcal D_k$9

and hence

$k^2$0

2. CR tournaments, basic tournaments, and strong CR tournaments

The paper introduces CR-associated vertices as the local mechanism governing admissible one-vertex extensions. For vertices $k^2$1 in a tournament $k^2$2, they are covertices if they have identical orientation to every other vertex, and revertices if they have opposite orientation to every other vertex. Formally, when $k^2$3,

$k^2$4

for covertices, and

$k^2$5

for revertices, where $k^2$6 if $k^2$7, and $k^2$8 otherwise. If two vertices are covertices or revertices, they are CR-associated (Zeng et al., 10 Aug 2025).

Let $k^2$9, and let $\mathcal D_k\setminus \mathcal D_{k-2}$0 specify the orientations between $\mathcal D_k\setminus \mathcal D_{k-2}$1 and $\mathcal D_k\setminus \mathcal D_{k-2}$2. Write $\mathcal D_k\setminus \mathcal D_{k-2}$3 for the enlarged tournament. Then $\mathcal D_k\setminus \mathcal D_{k-2}$4 is a CR vertex for $\mathcal D_k\setminus \mathcal D_{k-2}$5 with $\mathcal D_k\setminus \mathcal D_{k-2}$6 if in $\mathcal D_k\setminus \mathcal D_{k-2}$7 there exists some $\mathcal D_k\setminus \mathcal D_{k-2}$8 such that $\mathcal D_k\setminus \mathcal D_{k-2}$9 and $1$0 are CR-associated. Otherwise $1$1 is a non-CR vertex.

This leads to the main definitions. Let $1$2.

• If $1$3 is a $1$4-tournament, a $1$5-tournament, or a diamond, it is a trivial CR tournament.
• Otherwise, $1$6 is a CR tournament if for every non-CR way of adding a new vertex $1$7, the resulting tournament leaves $1$8:

$1$9

Equivalently, for $H$0,

$H$1

A tournament $H$2 of order $H$3 is basic if it has no pair of CR-associated vertices. A CR tournament $H$4 is a strong CR tournament if every $H$5-transitive blowup of $H$6 is also a CR tournament. A basic CR tournament is both basic and CR, and a basic strong CR tournament is both basic and strong CR.

These notions are switching invariant: CR-associated relation, being a CR tournament, being a strong CR tournament, and being basic are all invariant under switching, and hence under switching isomorphism. This makes switching equivalence the natural ambient equivalence relation for the theory.

3. Structural characterization of basic strong CR tournaments

The central theorem states that strong CR tournaments are precisely the templates whose transitive blowups exhaust the determinant layer inside their containment class. Let $H$7 denote the class of tournaments containing a subtournament switching isomorphic to $H$8. Then for odd $H$9 and a basic tournament $S_T$00, the following are equivalent:

1. $S_T$01 is a strong CR tournament;
2. all transitive blowups of $S_T$02 are CR tournaments;

3.

$S_T$03

This is the paper’s main characterization theorem (Zeng et al., 10 Aug 2025).

A basic strong CR tournament therefore acts as a rigid extremal template for the layer $S_T$04. If $S_T$05 contains a subtournament switching isomorphic to $S_T$06, then no additional structure beyond transitive blowup is permitted while staying inside the same determinant bound.

The basicness assumption is essential. If a tournament is not basic, then after switching it is a $S_T$07-transitive blowup of a smaller tournament. Thus basic tournaments are the irreducible cores of the theory. The paper also proves that in a basic tournament, a newly added CR vertex can be CR-associated with at most one original vertex, and if $S_T$08 is a transitive blowup of a basic tournament $S_T$09, then adding a non-CR vertex cannot accidentally produce something switching equivalent to a transitive blowup of $S_T$10. This suggests that the basic/strong CR distinction isolates the exact point at which blowup rigidity becomes canonical.

4. The canonical family $S_T$11 and the determinant layers $S_T$12

For $S_T$13, $S_T$14 is the $S_T$15-tournament with vertices $S_T$16 such that $S_T$17 is transitive with

$S_T$18

and $S_T$19 alternates against those vertices: $S_T$20 Thus $S_T$21 is transitive, $S_T$22 is a diamond, and for even $S_T$23,

$S_T$24

The paper proves two decisive facts: for even $S_T$25, $S_T$26 is a basic strong CR tournament; and all $S_T$27 are strong CR tournaments (Zeng et al., 10 Aug 2025). Even $S_T$28 are basic, while odd $S_T$29 are strong CR but not basic. This matters because even $S_T$30 provide the irreducible templates to which the main characterization theorem applies directly.

These results unify earlier classifications of low determinant layers. Existing results had shown that for $S_T$31, a tournament $S_T$32 is switching equivalent to a transitive blowup of $S_T$33. The strong CR theory yields a general higher-$S_T$34 statement with a containment condition: if $S_T$35 is odd and

$S_T$36

then

$S_T$37

This answers a question posed by Zeng and You.

The family $S_T$38 is therefore not merely a source of examples. It is the canonical source of strong CR templates, and it explains why determinant layers can be governed by blowups of a single tournament once a switching-isomorphic copy of that tournament is present.

5. Proof architecture and technical mechanisms

The proof theory of strong CR tournaments combines switching, transitive blowups, and determinant identities. One foundational lemma states that if $S_T$39 is a CR vertex for $S_T$40, then some switch of $S_T$41 is a $S_T$42-transitive blowup of $S_T$43. Since determinant and membership in $S_T$44 are invariant under switching and preserved under transitive blowups, this gives

$S_T$45

The difficult direction is to show that non-CR extensions force determinant $S_T$46.

A second recurring mechanism is the interaction between blowups and determinant multiplication. For a transitive tournament $S_T$47 of order $S_T$48,

$S_T$49

If a blowup $S_T$50 has some non-transitive part $S_T$51, then there is a subtournament with determinant $S_T$52; in the special case where one part is a $S_T$53-cycle and the others are singletons,

$S_T$54

This shows why transitive blowups are the only blowups compatible with staying inside a fixed determinant layer.

For the proof that even $S_T$55 are CR tournaments, the paper studies a non-CR extension $S_T$56 through its skew-adjacency matrix. A key determinant identity is

$S_T$57

A specialized corollary yields

$S_T$58

for suitable $S_T$59 and $S_T$60. This reduces determinant comparison to arithmetic control of sign patterns.

The most technical part introduces a combinatorial matrix $S_T$61 and row-sum parameters $S_T$62 such that

$S_T$63

Difference formulas for consecutive $S_T$64 then show that if the extension pattern is non-CR, one of these determinants must exceed $S_T$65. A plausible implication is that strong CR theory is driven less by global tournament structure than by an exact control of how sign-pattern perturbations propagate through skew-adjacency determinants.

6. Examples, neighboring usages of “CR,” and open problems

Several low-order examples anchor the theory. By definition, the $S_T$66-tournament, the $S_T$67-tournament, and the diamond are trivial CR tournaments. Every $S_T$68-tournament is a CR tournament. The paper also checks directly that $S_T$69 are strong CR tournaments for $S_T$70, and in particular $S_T$71 and $S_T$72 are basic strong CR tournaments (Zeng et al., 10 Aug 2025).

The paper does not provide an example of a CR tournament that is not strong CR. After examining low-order cases, the authors found none, and they raise this as an open question. The explicit open problems include:

1. Which CR tournaments are strong CR tournaments?
2. Is every CR tournament a strong CR tournament?
3. If not, find sufficient, necessary, or necessary-and-sufficient conditions for a CR tournament to be strong CR.
4. For $S_T$73, can one find finitely many basic tournaments $S_T$74 such that

$S_T$75

A common misconception is that “CR tournament” has a unique meaning across tournament theory. In fact, the same abbreviation appears elsewhere for the Condorcet Random model, a one-parameter upset model used in computational social choice. That literature does not define strong CR tournaments in the determinant-theoretic sense, and it treats “CR” as “Condorcet Random,” with upset probability $S_T$76 and $S_T$77 corresponding to the uniform random tournament (Mattei et al., 2016). Another nearby but distinct line studies strongly critical vertices in indecomposable tournaments, where “critical” and “strongly critical” refer to indecomposability rather than determinant layers; this is terminologically adjacent but mathematically separate (Rachid, 2023). This suggests that, in current usage, “strong CR tournament” is a specialized term of the determinant-and-switching framework, not a generic synonym for strong tournament structure.

In that framework, strong CR tournaments provide the rigid templates governing determinant-bounded classes through transitive blowups. Their defining feature is not strong connectivity, pancyclicity, or Erdős–Hajnal structure, but extension rigidity inside $S_T$78: once a basic strong CR tournament appears, the only way to remain in the same determinant layer is to enlarge it by transitive blowup and switching equivalence.