Impartial Hypergraph Games

We study two building games and two removing games played on a ﬁnite hypergraph. In each game two players take turns selecting vertices of the hypergraph until the set of jointly selected vertices satisﬁes a condition related to the edges of the hypergraph. The winner is the last player able to move. The building achievement game ends as soon as the set of selected vertices contains an edge. In the building avoidance game the players are not allowed to select a set that contains an edge. The removing achievement game ends as soon as the complement of the set of selected vertices no longer contains an edge. In the removing avoidance game the players are not allowed to select a set whose complement does not contain an edge. We develop some generic tools for ﬁnding the nim-value of these games and show that the nim-value can be an arbitrary nonnegative integer. The outcome of many of these games were previously determined for several special cases in algebraic and combinatorial settings. We provide several examples and show how our tools can be used to reﬁne these results by ﬁnding nim-values.


Introduction
Avoidance and achievement games, sometimes called positional games, are combinatorial games that are extensively studied in both impartial [20,21] and partizan [5] settings. The focus of this paper is a class of impartial games that provides a common framework for many of the special cases considered in the literature and listed in Section 8. This class of games is played on a finite hypergraph, so we call them impartial hypergraph games. Two players take turns selecting previously unselected vertices until the set of jointly selected vertices satisfies a condition related to the edges of the hypergraph. We consider four different games. Two of them are building games [1,3,13,14,36]. In these games, the players try to achieve or avoid having the set of selected vertices contain an edge. The other two games are removing games [23]. In these games, the players try to achieve or avoid having the complement of the set of selected vertices no longer contain an edge. In the removing games, the players essentially delete vertices until the remaining vertices no longer contain an edge. As is standard in normal play, the last player to move is the winner.
Every impartial game [2,32] has an associated nim-value or Sprague-Grundy number. The nim-value contains a lot of information about the game. It encodes the outcome of the game and describes the game's behavior with respect to game sums. Because of this, games with the same nim-value are considered equivalent.
Many hypergraph games can be effectively analyzed using structure theory. This theory uses an equivalence relation called structure equivalence on the game positions. Structure equivalence is compatible with the option structure of a hypergraph game. This means that the quotient digraph of the game digraph contains enough information to determine the nim-value of the game. This quotient digraph is often much smaller than the original digraph. So it provides a practical algorithm for finding the nim-value. It also provides a visualization of the game that can create useful insights for proving results about families of games. Structure equivalence is used for group and convex geometry games in [7,8,9,10,11,17,28].
After some background information on impartial games, hypergraphs, and closure systems in Section 2, we develop the basic general theory of impartial hypergraph games in Section 3. Structure theory is developed in Sections 4-6 as a generalization of results in [17]. In Section 7, we find the nim-values of games played on hypergaphs with relatively simple edge structures. Section 8 connects our theory to the existing literature on avoidance and achievement games. In Section 9, we show that the nim-value of a hypergraph game can be any nonnegative integer. We finish with a few open questions in Section 10.

Preliminaries
The parity of a non-negative integer n is denoted by pty(n) := n mod 2. The parity of a set A is the parity of the size of the set, that is, pty(A) := pty(|A|). For f : X → Y and A ⊆ X, we write f (A) := {f (a) | a ∈ A} for the image of the subset A. The complement of a set A is denoted by A ∁ . We use the notation ∁ V (H) = {V \ A | A ∈ H} or simply ∁(H) = {A ∁ | A ∈ H} for a family H of subsets of V .

Impartial games
We recall the basic terminology of impartial combinatorial games. See [2,32] for further details. An impartial game G consists of a finite set P of positions, a starting position, and a function Opt : P → 2 P that provides the set of options Opt(P ) for all P ∈ P. At the beginning of the game the starting position becomes the current position. Two players take turns picking an option of the current position to become the new current position. A position P is called terminal if it has no options, that is, Opt(P ) = ∅. The game ends when the current position becomes a terminal position. The winner of the game is the last player to move. Every game must finish after finitely many turns. An impartial game is essentially an acyclic digraph called the game digraph. The vertices are the positions of the game and the arrows connect positions to their options. Game play consists of moving a token along the arrows starting at the initial position until a sink is reached.
The minimum excludent mex(A) of a set A of nonnegative integers is the smallest nonnegative integer missing from A. The nim-value nim(P ) := mex(nim(Opt(P ))) of a position P is defined recursively as the minimum excludent of the nim-values of the options of P . The nim-value of a game is the nim-value of its starting position. A position P is losing (P -position) for the player about to move if nim(P ) = 0. A position P is winning (N -position) for the player about to move if nim(P ) ∕ = 0. The winning strategy is to move into a position with nim-value 0 if possible.
The sum of two impartial games G 1 and G 2 is an impartial game G 1 + G 2 , where in each turn the players make a valid move in exactly one of the two games. The positions of G 1 + G 2 are of the form (P, Q), where P is a position of G 1 and Q is a position of G 2 , and Opt(P, Q) = {P } × Opt(Q) ∪ Opt(P ) × {Q}.
Proposition 2.1. For all impartial games G, nim(G) = n if and only if G + * n is won by the second player.

Hypergraphs
A hypergraph H = (V, H) consist of a finite set V of vertices and a family H ⊆ 2 V of edges. Our general reference for hypergraphs is [12]. We do allow H to be empty or to contain the empty set. We say H is simple if H is a Sperner family. That is, no edge is contained in another edge.
A transversal of H is a subset T of V that intersects every edge. Transversals are also called hitting sets or vertex covers. The family Tr(H) of minimal transversals of H is a Sperner family. Finding Tr(H) is an important NP-complete problem [25,29] A set S of vertices of H is stable if S contains no edge of H. Stable sets are also called independent sets. We denote the family of maximal stable sets of H by S H . We also use the simpler notation S if the hypergraph H is clear from context. The following well-known fact plays an important role in our development.
This implies that P is maximal stable if and only if P ∁ is a minimal transversal of H. That is, If H is simple, then we also have H = Tr(∁(S H )), so the relationships can be summarized with the following diagram: S  Example 2.10. The family of maximal stable sets for the complete r-uniform hypergraph
A pre-closure operator on S is a function cl : 2 S → 2 S that is extensive and increasing. A subset P of S is dense or generating under a pre-closure operator cl if cl(P ) = S.
A closure system on a set S is a nonempty collection C of subsets of S that is closed under intersections. The empty intersection is allowed, so S = $ ∅ ∈ C. We say that the pair (S, C) or simply S is a closure space. We call the elements of C closed sets. A comprehensive reference for closure systems is the survey article [16].
Closure operators and closure systems are two sides of a coin. Given a closure operator cls : 2 S → 2 S , the range C = {cls(P ) | P ⊆ S} of the closure operator forms a closure system on S. Given a closure system C, the corresponding closure operator cls : 2 S → 2 S is defined by cls(P ) := $ {C ∈ C | P ⊆ C}. Example 2.13. Let G be a finite group. The collection C of subgroups of G is a closure system on G. The corresponding closure operator P + → 〈P 〉 : 2 G → 2 G outputs the subgroup 〈P 〉 generated by the subset P .

Impartial hypergraph games
Let H = (V, H) be a hypergraph. In an impartial hypergraph game on H, two players alternately select previously unselected vertices in V until the game ends at a terminal position. The last player to make a move wins the game. In all games the set P of jointly selected elements is the current position of the game. Observe that an option of a position always has the opposite parity. We consider two building games. The achieve game ACV(H) ends as soon as P contains one of the edges of H. In the avoid game AVD(H), the players are not allowed to select an element if the resulting P would contain one of the edges of H.
We also consider two removing games. The destroy game DST(H) ends as soon as P ∁ is a stable set of H. In the preserve game PRV(H), the players are not allowed to select an element if the resulting P ∁ would be a stable set of H.
The games ACV(H) and DST(H) are called achievement games, while AVD(H) and PRV(H) are called avoidance games. The terminology is summarized in the following table: achievement avoidance building achieve ACV avoid AVD removing destroy DST preserve PRV For a subset A of 2 V we define be the set of minimal elements of A with respect to inclusion. We also define For a fixed hypergaph game type, the three games played on (V, Min(H)), (V, H), and (V, Upp(H)) are the same. So we assume that a game is always played on a simple hypergraph H, so that H = Min(Upp(H)).  In fact, it is easy to verify using Proposition 2.1 that nim(ACV(H)) = pty(r) and nim(AVD(H)) = pty(r − 1) while nim(DST(H)) = pty(n − r + 1) and nim(PRV(H)) = pty(n − r). The winning strategy is simply random play.  The second player has a winning strategy for ACV(H) since nim(ACV(H)) = 0. The first player has a winning strategy for the other three games since nim(AVD(H)) = 2 = nim(DST(H)) and nim(PRV(H)) = 1. In fact, the first player wins these three games after one move. The hypergraph games for Tr(H) are the same. We will see later that this is no accident.
Note that the digraph of an avoidance game is a subdigraph of the corresponding achievement game. The positions missing from the avoidance game are exactly the terminal positions of the corresponding achievement game.
Also note that the nested sets {a, b} and {a, b, c} are both terminal positions of the achievement game ACV(H) = DST(Tr(H)). This can never happen for an avoidance game.
The option relationship for avoidance games is very simple. achievement avoidance ACV(H) DST(Tr(H)) The option relationship is slightly more complicated for the achievement games because terminal positions can be nested.   The options relations of the two games are also the same by Remark 3.5 since we already saw that the corresponding avoidance games are the same.
Remark 3.7. The underlying graph of the game digraph of an impartial hypergraph game is a subgraph of the graph constructed from the Hasse diagram of the Boolean lattice of subsets of V . The initial position is always the empty set.   Proof. First assume Q ∈ Opt(P ) in ACV(H). Then P is not a terminal position of ACV(H), and so P is a stable set of H. So P ∁ is not a game position of PRV(H). This means P ∁ cannot be an option of Q ∁ in PRV(H). Now assume P ∁ ∕ ∈ Opt(Q ∁ ) in PRV(H). Then P is a stable set of H. Hence P is a position of ACV(H) and P is not a terminal position. Thus Q ∈ Opt(P ) in ACV(H) by Remark 3.5.
The following is an easy consequence.     The next example shows that a game can be both an achivement and an avoidance game.
Proof. The backward direction clearly holds. Assume ACV(H) = AVD(K) and consider a nonempty position P of this game. Since P is stable in K, P \ {v} is also stable in K and hence a game position for all v ∈ P . Since P \ {v} is not a terminal position of ACV(H), P \ {v} ∪ {w} is also a game position for all w ∈ P ∁ . This shows that if Q is a subset of V satisfying |Q| " |P |, then Q is a game position. Let k be the size of the largest game position. Then every subset of V with size at most k is a game position and the terminal positions are the elements of The following result often provides a simple way to find the nim-value of an avoidance game.
Proposition 3.15. If every set in S H has the same parity r, then nim(AVD(H)) = r.
Proof. The sets in S H are the terminal positions of the game. The second player wins AVD(H) + * r using random play.
A version of this result is true for achievement games but it is not very useful since finding the terminal positions of an achievement game is often difficult.

Structure theory for the building games AVD and ACV
In this section we develop structure theory for hypergraph games. This is our main tool to find the nim-value of a hypergraph game. The idea of structure equivalence originates in [17]. Structure theory is successfully used in [7,8,9,10,11,28] to analyze group and convex geometry generating games.

Structure equivalence
We use this to define an equivalence relation on 2 V . Let be the closure system generated by the family S H of stable sets. The closure of a subset P of V in this closure system is denoted by ⌈P ⌉ := $ φ H (P ). Note that ⌈I⌉ = I for all I ∈ I H . In this closure system, H is the family of minimal generating sets and S H is the family of maximal non-generating sets.
The smallest set Φ H := $ S H in I H is called the Frattini subset. The Frattini subset is structure equivalent to the empty set. In fact, ⌈∅⌉ = Φ H .
The following is a generalization of [28,Proposition 4.9]. It shows that structure equivalence is the cospanning relation of the closure operator P + → ⌈P ⌉ as defined in [27,37]. Proof. The forward direction is clear from the definitions. For a contradiction suppose that ⌈P ⌉ = ⌈Q⌉ but there is an S ∈ S H such that P ⊆ S and Q ∕ ⊆ S. Then there is a The following result is easy to verify from the definitions but it also follows [37, Theorem 2.15, Proposition 3.4] from the fact that structure equivalence is the cospanning relation of a closure operator.
Consider a building hypergraph game on H = (V, H). We restrict structure equivalence to the set of game positions. The structure class X I for I ∈ I H consists of the game positions that are structure equivalent to I. Note that I is the largest element of X I for all I ∈ I H \ {V }. If P is a game position, then P ∼ ⌈P ⌉ and P ∈ X ⌈P ⌉ .
For ACV(H), the structure class X V contains all the terminal game positions. Note that V might not be a game position in which case V ∕ ∈ X V . The mapping I + → X I is a bijection from I H to the set of structure classes.
For AVD(H), no game position is structure equivalent to V so X V = ∅. The mapping I + → X I is a bijection from I H \ {V } to the set of nonempty structure classes.
A structure class is called terminal if it contains a terminal position. The only terminal structure class is X V for ACV(H). The terminal structure classes are X I with I ∈ S H for AVD(H).

Compatibility of game options with structure equivalence
Our goal is to show that structure equivalence is compatible with the option structure of the building hypergraph games. The following is a generalization of [17, Corollary 3.11].
It remains to show that Q ∪ {v} is a game position and an option of Q. First consider the case when G is an achievement game. Since P is not a terminal position, P must be stable. So P is contained in a maximal stable set S. Position Q is also contained in S, and so Q is also stable. Hence Q ∪ {v} must be an option of Q by Remark 3.5. Now consider the case when G is an avoidance game. Since P ∪ {r} is stable, the structure equivalent Q∪{r} is also stable. Hence Q∪{r} is an option of Q by Remark 3.4.
The following is a generalization of [17,Lemma 3.14].
The following is a generalization of [17,Proposition 3.15].
Proposition 4.8. Let P be the set of positions of a building hypergraph game G on H = (V, H). If P, Q ∈ P such that P ∼ Q and pty(P ) = pty(Q), then nim(P ) = nim(Q).
We say (P, Q) ≽ (M, N ) exactly when P ⊆ M and Q ⊆ N . Then (Z, ≽) is a finite partially ordered set. We proceed by structural induction on Z. Consider a minimal element (P, Q) of Z. If G = AVD(H), then we must have P = Q. If G = ACV(H) then P and Q might be different. For both games P and Q are terminal positions and so nim(P ) = 0 = nim(Q). So the claim holds for these minimal elements. Now let (P, Q) be an element of Z that is not minimal and let I := ⌈P ⌉ = ⌈Q⌉. Then I ∕ = V . We consider several cases. The claim clearly holds if P = I = Q.
Next, assume P ∕ = I ∕ = Q. Then both P and Q have options in X I . In fact, P ∪ {u} ∈ Opt(P ) ∩ X I for each u ∈ I \ P and Q ∪ {v} ∈ Opt(Q) ∩ X I for each v ∈ I \ Q. If

Type calculus
Consider a building game on the simple hypergraph H = (V, H). We call the recursive computation of types using the options of structure classes type calculus.
Note that X V is special since it only contains the terminal positions of the achieve game. The type of a structure class X I encodes the parity of I and the nim numbers of the positions in X I . Proposition 4.12. If P ∈ X I then nim(P ) = nim pty(P ) (X I ).
Proof. We use structural induction on the positions, together with Propositions 4.5 and 4.8. The statement is clearly true for I = V . Now assume that I ∕ = V .
Any option Q of position I is in X J for some X J ∈ Opt(X I ). On the other hand, if X J ∈ Opt(X I ) then X J contains an option Q of I. Hence nim(I) = mex(nim 1−pty(I) (Opt(I))) = nim pty(I) (X I ) by induction.
First assume that P is a position in X I such that pty(P ) = pty(I). Then nim(P ) = nim(I) by Proposition 4.8. Thus nim(P ) = nim(I) = nim pty(I) (X I ) = nim pty(P ) (X I ).
Now assume that P is a position in X I such that pty(P ) = 1−pty(I). Since P is strictly smaller than I and I ∕ = V , P must have an option in X I . Every Q in Opt(P ) ∩ X I satisfies nim(Q) = nim pty(Q) (X I ) = nim pty(I) (X I ) by induction. Proposition 4.5 implies that every option of P that is not in X I must be a position in X J for some X J ∈ Opt(X I ). On the other hand, Proposition 4.5 also implies that if X J ∈ Opt(X I ) then X J contains an option of P . Every Q in Opt(P ) ∩ X J with X J ∈ Opt(X I ) satisfies nim(Q) = nim pty(Q) (X J ) = nim pty(I) (X J ) by induction. Thus nim(P ) = mex(nim(Opt(P ))) = mex(nim((Opt(P ) ∩ & Opt(X I )) ∪ (Opt(P ) ∩ X I )) = mex(nim pty(I) (Opt(X I )) ∪ {nim pty(I) (X I )}) = nim 1−pty(I) (X I ) = nim pty(P ) (X I ).
Note that X I = {I} is possible. In this case X I contains no position with parity 1 − pty(I). Also note that the nim number of the game is the nim number nim(∅) = mex 0 (X Φ H ) of the starting position ∅.

Structure diagram
The structure digraph of a building game on the simple hypergraph H has vertex set {X I | I ∈ I H } and arrow set {(X I , X J ) | X J ∈ Opt(X I )}. We visualize the structure digraph with a structure diagram that also shows the type of each structure class. Within a structure diagram, a vertex X I is represented by a triangle pointing up or down depending on the parity of I. The triangle points down when pty(I) = 1 and points up when pty(I) = 0. The numbers within each triangle represent the nim numbers of the positions within the structure class. The first number is the common nim number nim 0 (X I ) of all even positions in X I , while the second number is the common nim number nim 1 (X I ) of all odd positions in X I . The algorithm is useful because instead of processing the potentially huge full game digraph we only need to process the structure digraph which is a hopefully smaller quotient digraph. Of course, the larger the structure equivalence classes are, the better the algorithm works.

Structure theory for the removing games PRV and DST
A removing game on a simple hypergraph H is equal to a building game on Tr(H). So for these games the appropriate structure equivalence is with respect to Tr(H). Note that S Tr(H) = ∁(Tr(Tr(H))) = ∁(H).

Simplified structure diagram
The structure diagram can be relatively easily found by a computer but it can be too large to provide any intuition about the game. So it is often useful to make further identifications in the structure diagram to build a simplified structure diagram. We want to make enough identifications to create a manageable diagram but too many identifications results in a simple but meaningless diagram. This is a delicate balance and the best approach depends on the hypergraph. The automorphism group of the hypergraph or of the structure diagram provides good opportunities for identifications but these automorphism groups can be difficult to compute. In this paper we use easy to compute conditions. We identify X I and X J if the following conditions hold.
1. pty(I) = pty(J); 2. type(Opt(X I )) = type(Opt(X J )); 3. The lengths of the longest directed paths starting at X I and at X J are the same.
The first two conditions guarantee that type(X I ) = type(X J ). The third condition avoids vertical collapsing. These conditions rely on the outgoing arrows of the structure digraph and ignore the incoming arrows. In the simplified structure diagram we use shaded triangles if they represent several structure classes. Note that we identified all the structure classes corresponding to the maximal stable sets even though X {2,4} is the only structure class that is an option of X Φ H .

Special hypergraphs
In this section we find criteria for the nim-value of games played on hypergraphs with a relatively simple structure. We are mainly concerned about the building games but many We warm up with a very simple case.

Single edge
In this subsection we study games played on a hypergraph with a single edge. The removing games are easy to analyze.

Pairwise disjoint edges
In this subsection we consider hypergraphs for which H contains pairwise disjoint sets. The achieve game can be quite complicated even in this simple case.

Pairwise disjoint maximal stable sets
In this subsection we consider hypergraphs for which S H contains pairwise disjoint sets.
One can now easily verify the claim by drawing the simplified structure diagrams. These diagrams are extensions of the diagrams with the additional structure class X V in Figure 7.2.

Examples
There is an extensive literature about games that are impartial hypergraph games. Many of these games [35] start with an empty graph on a set of vertices and the players build a graph by adding edges until the graph has a certain property. These include diameter avoidance games [13], triangle avoidance games [15,31], saturation games [33], path achievement games [36], connect-it games [23], minimum degree games [19], and degree games [22].
Other games start with a graph and the players select vertices until the set of selected vertices has a certain property. These include geodetic closure games [6,14,24,30] and general position games [26,34].
In this section we provide some detailed examples to show how these games can be considered as hypergraph games.

Generate and do not generate games
Some examples in this section rely on the notion of a generating set with respect to a pre-closure operator. Given a pre-closure operator cl : 2 S → 2 S and a subset W of 2 S of winning sets. Let H := Min({P ⊆ S | W ⊆ cl(P ) for some W ∈ W}). The achievement game ACV(H) and avoidance game AVD(H) on the hypergraph H := (S, H) are often called generate and do not generate and denoted by GEN(cls, W) and DNG(cls, W). We can think of the elements of H as the minimal generating sets while the stable sets as maximal non-generating sets.
The family W of winning sets is most often only contains S. In this case H is the set of dense subsets of S and the simplified notations GEN(cl) and DNG(cl) can be used. We even use the notations GEN(S) and DNG(S) if S has a default standard closure operator. We call removing games in this context terminate TER(cl) and do not terminate DNT(cl). They are studied in [6] and in [23] under the name disconnect-it games.

Group generating games
The generate and do not generate games on groups are studied in [3,4,17,8,9,10,7,11]. The closure operator is P + → 〈P 〉 : 2 G → 2 G is the generated subgroup operator on a finite group G. The family of winning sets is W = {G}.

Convex closure games
The generate game on convex geometries is studied in [28] using the abstract convex closure operator τ : 2 S → 2 S on a finite point set S.

Geodetic closure games
Let G = (V, E) be a connected graph. A geodesic is a shortest path between two vertices. The geodetic closure of a subset P of V is the set (P ) of vertices that are contained on a geodesic between two vertices of P . The mapping P + → (P ) : 2 V → 2 V is a pre-closure operator. A set P of vertices is called a geodetic cover if (P ) = V . A geodetic basis is a geodetic cover with minimum size. The outcome for the achievement and avoidance games for this pre-closure operator are studied in [14,24,30]. We can easily recover the following results of [24].

Minimum degree games for graphs
Avoidance and achievement games with the purpose of making a graph with n vertices and a minimum degree δ are studied in [19]. We can generalize these games for a given connected graph G = (V, E). Consider the hypergraph H = (E, H) where H is the family consisting of each minimal set A of edges of G for which each vertex of G is incident to at least δ edges in A.

Degree achievement and avoidance games for graphs
Avoidance and achievement games with the purpose of making a graph with n vertices and a vertex with a given degree d are studied in [22]. We demonstrate these games on an example.

Spectrum of nim-values
Our goal in this section is to show that the nim-value of AVD(H) and ACV(H) can be any nonnegative integer. To do this, we need to develop some tools to construct structure diagrams for games created from simpler games. These tools are interesting in their own right.  Since we need to deal with games on several hypergraphs, We will use the notations Opt H , type H , X H I to indicate the hypergraph H. Proposition 9.2. Consider the avoid game on the simple hypergraphs H = (V, H) and

Nim-values of AVD
Proof. Part (1) follows from the computation Parts (2-4) are immediate consequences. It is clear that pty(I ∪ {w}) = 1 − pty(I). The rest of Part (5) follows by structural induction on the structure classes.
The result tells us how to construct the structure diagram of AVD(K) from the structure diagram of AVD(H). We need to flip the parity and swap the nim-values for each structure class. We demonstrate this on an example.  We are now ready to construct a family H k of hypergraphs satisfying nim(AVD(H K )) = k for each nonnegative integer k. The construction is done through the family of maximal stable sets. S ∨ k−1 , k is odd S ∨ k−2 ⊎ S ∨ k−1 , k is even recursively. We also let H k := (V k , H k ) with V k := # S k and H k := Tr(∁ V k (S k )). Example 9.9. Figure 9.3 shows the structure diagrams of AVD(H k ) for k ∈ {0, . . . , 5}. Note that in every structure diagram X Φ has every other structure class as an option. Claim (2) follows from the computation {(1, 2l + 1, 2l) | 0 " 2l " (k + 1) − 1}.
Corollary 9.11. The nim-value of AVD(H) can be any nonnegative integer.

Nim-values of ACV
Our goal in this section is to show that the nim-value of ACV(H) can also be any nonnegative integer. Proof. Let w ∈ V \ # S H . Then {w} ∈ X V for ACV(H). Hence X V ∈ Opt(X I ) for all I ∈ I H \ V since I ∪ {w} ∈ X V . Thus the structure digraph of ACV(H) can be constructed from the structure digraph of AVD(H) by connecting every structure class to the additional structure class X V . Structural induction on the structure classes combined with type calculus shows that the nim-values in type(X I ) are one larger for ACV(H) than they were for AVD(H).
Corollary 9.14. The nim-value of ACV(H) can be any nonnegative integer.
Proof. Figure 4.1 shows that the nim-value of ACV(H) can be 0. We insert an additional vertex w into V k without altering S k in Definition 9.6. This creates the hypergraphH k = (Ṽ k , Tr(∁Ṽ k (S k ))) withṼ k = V k ∪{w}. The avoidance game remains the same since marking this vertex is never allowed in AVD(H k ). Thus nim(ACV(H k )) = nim(AVD(H k )) + 1 = k + 1.

Nim-values of PRV and DST
Since the removing games are actually building games in disguise, we have the following.

Further questions
We finish with a few unresolved questions for further study.
1. We saw in Example 3.8 that not every impartial game can be realized as a hypergraph game. What properties must a digraph have to be the game digraph of a hypergraph game? 2. How to use information about the automorphism group of the hypergraph together with the structure diagram? A simplified structure diagram takes advantage of certain symmetries of the hypergraph but only after the structure diagram is built.
Using the automorphism group in Algorithm 4.13 might speed up the computations. 3. Proposition 3.10 shows that the digraphs of ACV(H) and PRV(H) are complementary. Does this connection force any restriction on the nim-values of these games? 4. Hypergraphs can have many interesting properties. How do these properties translate into results about the nim-value of the corresponding hypergraph games? 5. The nim-value of a group generating game cannot be an arbitrary nonnegative integer [11]. This is a consequence of Lagrange's Theorem, which restricts the allowed families of maximal subgroups. Can we determine the spectrum of nimvalues for the other games mentioned in Section 8.