CR Tournaments and Determinantal Switching

• CR Tournaments are defined by the behavior of determinants of skew-adjacency matrices, emphasizing structured vertex additions and precise determinant preservation.
• They utilize the concepts of covertices and revertices to categorize tournaments within determinant classes, ensuring that nonconforming extensions leave specific determinant bounds.
• Switching equivalence and transitive blowup methods are central to characterizing basic and strong CR tournaments, providing a clear structural framework in Dk classes.

CR tournaments are tournaments defined through the behavior of determinants of skew-adjacency matrices under vertex addition. The notion arises in the study of the classes $\mathcal{D}_k$, where $\mathcal{D}_k$ consists of tournaments all of whose subtournaments have determinant at most $k^2$, and it isolates tournaments with the special property that adding any vertex that does not conform to their structure increases the maximum value of determinants among their subtournaments. In this setting, CR tournaments, strong CR tournaments, and basic tournaments provide a structural language for describing when membership in $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$ is preserved exactly by transitive blowups and switching equivalence (Zeng et al., 10 Aug 2025).

1. Determinantal framework and the classes $\mathcal{D}_k$

Let $T$ be a tournament on vertices $v_1,\dots,v_n$. Its adjacency matrix is

$$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$

and its skew-adjacency matrix is

$S_T=A_T-A_T^{\mathsf T}.$

The determinant of the tournament is defined by

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

This determinant is independent of the chosen vertex ordering. By Pfaffian theory for skew-symmetric matrices, $\mathcal{D}_k$0 for odd-order tournaments, and for even order the determinant is the square of an odd integer (Zeng et al., 10 Aug 2025).

For a positive odd integer $\mathcal{D}_k$1,

$\mathcal{D}_k$2

Equivalently, $\mathcal{D}_k$3 iff all principal minors of $\mathcal{D}_k$4 are $\mathcal{D}_k$5. The set $\mathcal{D}_k$6 is closed under switching. For odd $\mathcal{D}_k$7, the condition

$\mathcal{D}_k$8

means that all subtournaments of $\mathcal{D}_k$9 have determinant at most $k^2$0, at least one subtournament has determinant exactly $k^2$1, and no smaller bound $k^2$2 suffices (Zeng et al., 10 Aug 2025).

Earlier results summarized in the paper show that, for $k^2$3, a tournament $k^2$4 if and only if $k^2$5 is switching equivalent to a transitive blowup of $k^2$6. CR tournaments generalize the structural mechanism behind this phenomenon by identifying tournaments for which any nonconforming one-vertex extension necessarily leaves $k^2$7 (Zeng et al., 10 Aug 2025).

2. CR-associated vertices and the definition of CR tournaments

The local relations underlying CR tournaments are formulated through covertices and revertices. For a tournament $k^2$8, and vertices $k^2$9, if $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$0 then they are called covertices and revertices by convention. If $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$1, then $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$2 are covertices if for every $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$3,

$\mathcal{D}_k\setminus\mathcal{D}_{k-2}$4

and they are revertices if for every $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$5,

$\mathcal{D}_k\setminus\mathcal{D}_{k-2}$6

where $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$7 if $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$8, and $\mathcal{D}_k\setminus\mathcal{D}_{k-2}$9 if $\mathcal{D}_k$0. If two vertices are either covertices or revertices, they are called CR-associated vertices (Zeng et al., 10 Aug 2025).

A useful equivalent criterion is that $\mathcal{D}_k$1 are covertices iff

$\mathcal{D}_k$2

for all $\mathcal{D}_k$3, and are revertices iff

$\mathcal{D}_k$4

for all such $\mathcal{D}_k$5. These relations encode the two extremal ways a new vertex can imitate or invert the orientation pattern of an existing one (Zeng et al., 10 Aug 2025).

Let $\mathcal{D}_k$6, let $\mathcal{D}_k$7 be a dominating relation between $\mathcal{D}_k$8 and $\mathcal{D}_k$9, and denote the resulting tournament by $T$0. The vertex $T$1 is a CR vertex for $T$2 with $T$3 if in $T$4 there exists some $T$5 such that $T$6 and $T$7 are CR-associated; otherwise $T$8 is a non-CR vertex for $T$9 with $v_1,\dots,v_n$0. The key preservation lemma states that if

$v_1,\dots,v_n$1

and $v_1,\dots,v_n$2 is a CR vertex for $v_1,\dots,v_n$3 with $v_1,\dots,v_n$4, then

$v_1,\dots,v_n$5

The paper interprets this via switching: if $v_1,\dots,v_n$6 is CR, then by switching one can turn $v_1,\dots,v_n$7 into a 1-transitive blowup of $v_1,\dots,v_n$8, and transitive blowups preserve membership in $v_1,\dots,v_n$9 (Zeng et al., 10 Aug 2025).

This leads to the definition of a CR tournament. If $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$0 is a $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$1-tournament, a $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$2-tournament, or a diamond, then $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$3 is called a trivial CR tournament. If $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$4 is not trivial, then $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$5 is a CR tournament if for every dominating relation $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$6 such that $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$7 is a non-CR vertex for $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$8 with $$A_T=[a_{ij}],\qquad a_{ij}=1 \text{ if } v_i\to v_j,\ \ 0 \text{ otherwise},$$9, one has

$S_T=A_T-A_T^{\mathsf T}.$0

Equivalently,

$S_T=A_T-A_T^{\mathsf T}.$1

for every $S_T=A_T-A_T^{\mathsf T}.$2 and every dominating relation $S_T=A_T-A_T^{\mathsf T}.$3 (Zeng et al., 10 Aug 2025).

3. Basic tournaments, strong CR tournaments, and switching invariance

A tournament $S_T=A_T-A_T^{\mathsf T}.$4 of order $S_T=A_T-A_T^{\mathsf T}.$5 is basic if there do not exist two vertices that are CR-associated in $S_T=A_T-A_T^{\mathsf T}.$6. The paper motivates this as a notion of basic structure: if CR-associated vertices exist, then the tournament can be viewed as a 1-transitive blowup of a smaller tournament after switching. A CR tournament is called basic CR if it is both basic and CR (Zeng et al., 10 Aug 2025).

A CR tournament is called strong CR if every 1-transitive blowup of it is also CR. It is called basic strong CR if it is both basic and strong CR. These refinements are designed to separate the local rigidity condition encoded by CR from the global closure property under repeated transitive blowup (Zeng et al., 10 Aug 2025).

Switching is structurally central throughout the theory. The paper proves that switching preserves being CR, being strong CR, being basic, being basic CR, and being basic strong CR. It also proves that if a tournament contains CR-associated vertices, then after switching it is a 1-transitive blowup of a smaller tournament. This suggests that the basic tournaments serve as the fundamental templates from which the non-basic members of $S_T=A_T-A_T^{\mathsf T}.$7 are generated (Zeng et al., 10 Aug 2025).

The distinction between CR and strong CR is substantive. CR controls one-step vertex addition, while strong CR requires that every 1-transitive blowup remain within the same rigidity class. The paper closes by asking whether every CR tournament is strong CR, indicating that the separation between the two notions is not yet resolved (Zeng et al., 10 Aug 2025).

4. Structural theorem for basic strong CR tournaments

The central theorem concerns a basic tournament

$S_T=A_T-A_T^{\mathsf T}.$8

with $S_T=A_T-A_T^{\mathsf T}.$9 odd. Let $\det(T):=\det(S_T).$0 denote the class of tournaments that contain a subtournament switching isomorphic to $\det(T):=\det(S_T).$1. Then the following are equivalent:

1. $\det(T):=\det(S_T).$2 is a strong CR tournament.
2. All transitive blowups of $\det(T):=\det(S_T).$3 are CR tournaments.
3. For every tournament $\det(T):=\det(S_T).$4,

$\det(T):=\det(S_T).$5

This is Theorem 15 in the paper and it is the main abstract characterization of strong CR behavior (Zeng et al., 10 Aug 2025).

The theorem converts a local extension property into a global recognition criterion. Once $\det(T):=\det(S_T).$6 is basic strong CR, the entire subclass of tournaments in $\det(T):=\det(S_T).$7 that contain $\det(T):=\det(S_T).$8 is exactly the switching-equivalence closure of the transitive blowups of $\det(T):=\det(S_T).$9. No additional configurations occur. In this sense, a basic strong CR tournament functions as a complete structural seed for its ambient determinant class (Zeng et al., 10 Aug 2025).

A key intermediate fact is Lemma 13: if $\mathcal{D}_k$00 is basic and $\mathcal{D}_k$01 is a non-CR vertex for a generated tournament $\mathcal{D}_k$02, then $\mathcal{D}_k$03 can be CR-associated with at most one vertex of $\mathcal{D}_k$04. The proof of Theorem 15 then partitions the outside vertices according to coversion and reversion with respect to a fixed copy of $\mathcal{D}_k$05, uses switching to normalize orientations, and shows that each part must be transitive; otherwise one obtains a non-CR extension that remains in $\mathcal{D}_k$06, contradicting the defining property (Zeng et al., 10 Aug 2025).

This theorem is the main reason strong CR tournaments matter. It shows that the determinant constraint defining $\mathcal{D}_k$07 can, in favorable cases, be expressed purely as a blowup classification relative to a single basic model $\mathcal{D}_k$08. That mechanism is what the paper then establishes concretely for the family $\mathcal{D}_k$09 (Zeng et al., 10 Aug 2025).

5. The family $\mathcal{D}_k$10 and the classification of $\mathcal{D}_k$11

For $\mathcal{D}_k$12, $\mathcal{D}_k$13 is the tournament on vertices $\mathcal{D}_k$14 such that $\mathcal{D}_k$15 is transitive with

$\mathcal{D}_k$16

and $\mathcal{D}_k$17 dominates the odd-indexed vertices and is dominated by the even-indexed ones: $\mathcal{D}_k$18 Thus $\mathcal{D}_k$19 is transitive, and $\mathcal{D}_k$20 is a diamond (Zeng et al., 10 Aug 2025).

The family $\mathcal{D}_k$21 is determinant-extremal in a precise sense. The paper recalls that

$\mathcal{D}_k$22

for even $\mathcal{D}_k$23. It then proves two foundational theorems: if $\mathcal{D}_k$24 is even, then $\mathcal{D}_k$25 is a basic strong CR tournament; and, more generally, all $\mathcal{D}_k$26 are strong CR tournaments (Zeng et al., 10 Aug 2025).

The proof for even $\mathcal{D}_k$27 is the technical core of the paper. It introduces a refined encoding $\mathcal{D}_k$28 of the dominating relation of a new vertex, analyzes the number $\mathcal{D}_k$29 of blocks in this encoding, and, in the setting $\mathcal{D}_k$30, identifies the CR case with $\mathcal{D}_k$31; for $\mathcal{D}_k$32, the non-CR case corresponds to $\mathcal{D}_k$33. The argument then uses a $\mathcal{D}_k$34-matrix $\mathcal{D}_k$35, its diagonal vectors $\mathcal{D}_k$36, and the determinant identity

$\mathcal{D}_k$37

for a skew-symmetric matrix with a transitive block. This machinery converts the combinatorics of the extension pattern into explicit determinant inequalities, forcing non-CR extensions of $\mathcal{D}_k$38 to leave $\mathcal{D}_k$39 (Zeng et al., 10 Aug 2025).

These results answer a question posed by Zeng and You. For odd $\mathcal{D}_k$40, if

$\mathcal{D}_k$41

then

$\mathcal{D}_k$42

Thus, among tournaments already known to lie at determinant level $\mathcal{D}_k$43, the additional condition of containing a subtournament switching isomorphic to $\mathcal{D}_k$44 is exactly what characterizes the blowup class of $\mathcal{D}_k$45 (Zeng et al., 10 Aug 2025).

6. Terminological scope, misconceptions, and open problems

The initials “CR” are overloaded in tournament theory. In the determinant-based sense discussed here, CR tournaments are the objects introduced in “CR tournaments” (Zeng et al., 10 Aug 2025). In other literatures, CR denotes the Condorcet Random tournament model used in empirical studies of single-elimination data (Mattei et al., 2016), and it also denotes the Catch-Up Rule in service sports (Brams et al., 2018). The determinant-based CR theory is distinct from both usages.

Within the determinant framework, the paper leaves several open questions. It asks which CR tournaments are strong CR tournaments, whether every CR tournament is strong CR, and, if not, what necessary or sufficient conditions characterize the strong CR subclass. It also asks whether, for odd $\mathcal{D}_k$46, there exist finitely many basic tournaments $\mathcal{D}_k$47 such that

$\mathcal{D}_k$48

This is presented as a broad generalization of the known $\mathcal{D}_k$49 and $\mathcal{D}_k$50 characterizations (Zeng et al., 10 Aug 2025).

The present state of the subject therefore combines a sharp abstract theorem with a concrete model family. Basic strong CR tournaments furnish exact blowup classifications inside $\mathcal{D}_k$51, and the family $\mathcal{D}_k$52 supplies the first fully developed example. A plausible implication is that further progress on CR tournaments will come from identifying additional basic strong CR templates and determining whether the determinant-level classes $\mathcal{D}_k$53 admit finite template descriptions beyond the $\mathcal{D}_k$54 family (Zeng et al., 10 Aug 2025).