The Reflexive Dimension of (0,1)-Polytopes

THE REFLEXIVE DIMENSION OF (0,1)-POLYTOPES

AKIYOSHI TSUCHIYA

ABSTRACT. Haase and Melnikov showed that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. The reflexive dimension of a lattice polytope P is the minimal d so that P is unimodularly equivalent to a face of some d- dimensional reflexive polytope. Computing the reflexive dimension of a lattice polytope is a hard problem in general. In this survey, we discuss the reflexive dimension of a (0,1)- polytope. In particular, virtue of the algebraic technique on Gro¨bner bases and a linear algebraic technique, many families of reflexive polytopes arising from several classes of (0,1)-polytopes are presented, and we see that the (0,1)-polytopes are unimodularly equivalent to facets of some reflexive polytopes.

INTRODUCTION The reflexive polytope is one of the keywords belonging to the current trends in the research of convex polytopes. In fact, many authors have studied reflexive polytopes from the viewpoints of combinatorics, commutative algebra and algebraic geometry. Hence, finding new classes of reflexive polytopes is an important problem. A lattice polytope is a convex polytope all of whose vertices have integer coordinates. d d Two lattice polytopes P R and P′ R ′ are said to be unimodularly equivalent if ⊆ ⊆ there exists an affine map from the affine span aff(P) of P to the affine span aff(P′) of d d P′ that maps Z aff(P) bijectively onto Z ′ aff(P′) and that maps P to P′. Note ∩ ∩ that every lattice polytope is unimodularly equivalent to a full-dimensional one. A lattice polytope P Rd of dimension d is called reflexive if the origin of Rd is the unique lattice ⊂ point belonging to the interior of P and its dual polytope d P∨ := y R x,y 1 for all x P { ∈ | 〈〉 ≤ ∈ } is also a lattice polytope, where x,y is the usual inner product of Rd. It is known that 〈〉 reflexive polytopes correspond to Gorenstein toric Fano varieties, and they are related to mirror symmetry (see, e.g., [1, 3]). In each dimension, there exist only finitely many reflexive polytopes up to unimodular equivalence ([15]). Moreover, Haase and Melnikov showed that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope ([4]). From this result they defined the reflexive dimension of a lattice polytope.

2010 Mathematics Subject Classification. 13P10, 52B12, 52B20. Key words and phrases. reflexive polytope, reflexive dimension, edge polytope, order polytope, chain polytope, stable set polytope, perfect graph, normal polytope, (0,1)-polytope, integer decomposition property. 1 Definition 0.1. Let P be a lattice polytope. The reflexive dimension of P, denoted by rdim(P), is the minimal integer d such that P is unimodularly equivalent to a face of some d-dimensional reflexive polytope.

We immediately know the following proposition from the fact that there are only ﬁnitely many reﬂexive polytopes up to unimodular equivalence.

Proposition 0.2. Given a positive integer d, there exist up to unimodular equivalence only ﬁnitely many lattice polytopes whose reﬂexive dimensions are equal to d.

Our interest is to classify lattice polytopes whose reflexive dimensions are a given integer. We remark that classifying lattice polytopes whose reflexive dimensions are equal to their dimensions is equivalent to classifying reflexive polytopes. In particular, it is known that there is one reflexive polytope in dimension one, there are 16 in dimension two, 4319 in dimension three and 473800776 in dimension four according to computations by Kreuzer and Skarke [14]. As a next step, we focus on lattice polytopes whose reflexive dimensions are equal to their dimensions plus one. Namely, we consider the following question.

Question 0.3. For which lattice polytope P, does it follow that rdim(P) = dim(P)+1? We note that for a lattice polytope P, rdim(P) = dim(P) + 1 if and only if P is not reflexive and P is a facet of some reflexive polytope. As an example of Question 0.3, the reflexive dimensions of a d-dimensional unit cube and a d-dimensional unit simplex are d + 1. A stronger question is the following.

Question 0.4. For every (0,1)-polytope P, does it follow that rdim(P) = dim(P)+1? Equivalently, is every (0,1)-polytope a facet of some reﬂexive polytope?

In order to show that Question 0.4 has a positive answer for some class of (0,1)- polytopes, we give higher-dimensional construction of lattice polytopes. Given two lattice polytopes P Rd and Q Rd, we set the lattice polytope Ω(P,Q) Rd+1 with ⊂ ⊂ ⊂ Ω(P,Q) := conv (P 1 ) ( Q 1 ) . { × { } ∪ − × {− } } If P = Q, then we will write Ω(P) := Ω(P,P). We remark that the origin of Rd+1 is always a relative interior lattice point of Ω(P). Assume that P is full-dimensional. Then Ω(P) is also full-dimensional. In particular, P 1 is a facet of Ω(P). Hence × { } for a lattice polytope P Rd of dimension d, if Ω(P) is reflexive, then we know that ⊂ P is unimodularly equivalent to a facet of some reflexive polytope. In this survey, by using this construction, we will show that for several classes of (0,1)- polytopes, Question 0.4 has a positive answer. In particular, we will present many large families of reflexive polytopes arising from well-known classes of (0,1)-polytopes. This survey is organized as follows: In Section 1, we will introduce an algebraic technique to show that a given lattice polytope is reflexive. In particular, we will recall basic 2 materials and notation on toric ideals. In Section 2, we will give two families of reflexive polytopes arising from the order polytopes and the chain polytopes of finite partially ordered sets. In Section 3, we will give a family of reflexive polytopes arising from the stable set polytopes of perfect graphs. Finally, in Section 4, we will introduce a linear algebraic technique to show that a lattice polytope is reflexive, and we will give a family of reflexive polytopes arising from the edge polytopes of finite simple graphs.

Acknowledgment. The author is very grateful to the anonymous referee for his or her insightful report that led to signiﬁcant improvements of the form of the paper. This man- uscript was prepared as a contribution to the conference proceedings of the Interactions with Lattice Polytopes at Otto-von-Guericke-Universita¨t Magdeburg on September 14th - 16th, 2017. The author thanks the organizers, Christopher Borger, Alexander Kasprzyk, Benjamin Nill and Johannes Hofscheier, for their support during this conference. The author was partially supported by Grant-in-Aid for JSPS Fellows 16J01549.

1. TORIC IDEALS AND REFLEXIVE POLYTOPES In this section, we introduce an algebraic technique to show that a lattice polytope is 1 reflexive. First, we recall basic materials and notation on toric ideals. Let K[t± ,s] := 1 1 K[t1± ,...,td± ,s] be the Laurent polynomial ring in d + 1 variables over a field K. If d a a1 ad 1 d a = (a1,...,ad) Z , then t s is the Laurent monomial t1 td s K[t± ,s]. Let P R ∈ d ··· ∈ ⊂ be a lattice polytope of dimension d and P Z = a1,...,an . Then, the toric ring of 1 ∩ { a }a P is the subalgebra K[P] of K[t± ,s] generated by t 1 s,...,t n s over K. We regard {a } K[P] as a homogeneous algebra by setting each deg t i s = 1. Let K[x] := K[x1,...,xn] denote the polynomial ring in n variables over K. The toric ideal IP of P is the kernel of the surjective homomorphism π : K[x] K[P] defined by π(x ) = tai s for 1 i n. → i ≤ ≤ It is known that IP is generated by homogeneous binomials. See, e.g., [27]. Let < be a monomial order on K[x] and in<(IP ) the initial ideal of IP with respect to <. The initial ideal in<(IP ) is called squarefree if in<(IP ) is generated by squarefree monomi- als. The reverse lexicographic order on K[x] induced by the ordering x < < x n rev ··· rev 1 is the total order

Lemma 1.1 ([8, Lemma 1.1]). Let P Rd be a lattice polytope of dimension d such that d ⊂ d the origin 0 of R is contained in its interior and P Z = a1,...,an . Suppose that any ∩ { } lattice point in Zd+1 is a linear integer combination of the lattice points in P 1 and × { } there exists an ordering of the variables x < < x for which a = 0 such that i1 rev ··· rev in i1 the initial ideal in

Example 1.2. Set ai = ei for i = 1,...,d, ad+1 = e1 ed and ad+2 = 0, where − −d ··· − d e1,...,ed are the standard coordinate unit vectors of R . Let P R be the lattice d ⊂ polytope with P Z = a1,...,ad+2 . Then one has ∩ { } I = (x x xd+1). P 1 ··· d+1 − d+2

Hence the initial ideal in

2. REFLEXIVE POLYTOPES ARISING FROM ORDER POLYTOPES AND CHAIN POLYTOPES In this section, we give two families of reflexive polytopes with regular unimodular triangulations arising from order polytopes and chain polytopes of finite partially ordered sets. In particular, we show that Question 0.4 has a positive answer for order and for chain polytopes. First, we recall some terminologies of finite partially ordered sets and introduce two lattice polytopes arising from finite partially ordered sets. Let P denote a finite partially ordered set (poset, for short) on the ground set [d] := 1,...,d . A subset I of [d] is called { } a poset ideal of P if i I and j P together with j i in P, then j I. Note that the empty ∈ ∈ ≤ ∈ set 0/ and [d] are poset ideals of P. Let J (P) denote the set of poset ideals of P. A subset 4 A of [d] is called an antichain of P if i and j belonging to A with i = j are incomparable. ∕ In particular, the empty set 0/ and each 1-element subsets j are antichains of P. Let { } A (P) denote the set of antichains of P. For a poset ideal I of P, we write max(I) for the set of maximal elements of I. In particular, max(I) is an antichain. A linear extension of P is a permutation σ = i i i of [d] which satisfies a < b if i < i in P. 1 2 ··· d a b Stanley [26] introduced two classes of lattice polytopes arising from finite posets, which are called order polytopes and chain polytopes. The order polytope OP of P is d defined to be the convex polytope consisting of those (x1,...,xd) R such that ∈ (1) 0 x 1 for 1 i d; ≤ i ≤ ≤ ≤ (2) x x if i j in P. i ≥ j ≤ The chain polytope C is defined to be the convex polytope consisting of those (x ,...,x ) P 1 d ∈ Rd such that (1) x 0 for 1 i d; i ≥ ≤ ≤ (2) x + + x 1 for every maximal chain i < < i of P. i1 ··· ik ≤ 1 ··· k For each subset I [d], we define the (0,1)-vectors ρ(I) := ∑i I ei. In particular ρ(0/) ⊂ ∈ is the origin 0 of Rd. Both order polytopes and chain polytopes are (0,1)-polytopes of dimension d. In fact, in [26, Corollary 1.3 and Theorem 2.2], it is shown that the set of vertices of O = ρ(I) : I J (P) , { P} { ∈ } the set of vertices of C = ρ(A) : A A (P) . { P} { ∈ } Moreover, both order polytopes and chain polytopes are compressed, hence, they possess the integer decomposition property. However, the class of order polytopes is different from the class of chain polytopes ([6, Example 3.5 and Corollary 3.9]). Now, we consider the two lattice polytopes Ω(OP) and Ω(CP) for a finite poset P on [d]. In particular, we will see that the toric ideals of these lattice polytopes are squarefree with respect to some monomial orders on K[x]. Let

K[O] := K[ xI,yI I J (P) z ], { } ∈ ∪ { } K[C ] := K[ xmax(I),ymax(I) I J (P) z ] { } ∈ ∪ { } denote the polynomial rings over K, and deﬁne the surjective ring homomorphisms πO and πC by the following: πO : K[O] K[Ω(OP)] by setting • →ρ(I d+1 ) ρ(J d+1 ) πO (xI) = t ∪{ } s, πO (yJ) = t− ∪{ } s and πO (z) = s, πC : K[C ] K[Ω(CP)] by setting • → ρ(max(I) d+1 ) ρ(max(J) d+1 ) πC (xmax(I)) = t ∪{ } s, πC (ymax(J)) = t− ∪{ } s and πC (z) = s where I,J J (P). Then the toric ideal I (resp. I ) is the kernel of π (resp. ∈ Ω(OP) Ω(CP O πC ). Next, we introduce monomial orders

(resp. GC ) is a Gro¨bner basis of IΩ(OP) (resp. IΩ(CP)) with respect to

Theorem 2.2 ([11, Theorem 1.3]). Let P be a ﬁnite poset on [d]. Then each of Ω(OP) and Ω(CP) is a reﬂexive polytope with a regular unimodular triangulation. Hence we can obtain the following corollary.

Corollary 2.3. Let P be a ﬁnite poset on [d]. Then one has

rdim(OP) = rdim(CP) = d + 1.

Remark 2.4. In [11], larger families of reflexive polytopes are given. In fact, for two finite posets P and Q with P = Q = d, the three lattice polytopes Ω(O ,O ), Ω(O ,C ) and | | | | P Q P Q Ω(CP,CQ) are studied. By the same technique, we know that Ω(OP,OQ) is a reflexive polytope (with a regular unimodular triangulation) if and only if P and Q have a common linear extension, and Ω(OP,CQ) and Ω(CP,CQ) are always reflexive polytopes with regular unimodular triangulations ([11, Theorem 1.3]). 6 Remark 2.5. In [7, 9, 10], by using other constructions of lattice polytopes, large families of reflexive polytopes are presented. Given two lattice polytopes P Rd and Q Rd, ⊂ ⊂ we set the lattice polytope Γ(P,Q) Rd with ⊂ Γ(P,Q) := conv P ( Q) . { ∪ − } If P = Q, then we will write Γ(P) := Γ(P,P). The two lattice polytopes Ω(P,Q) and Γ(P,Q) often have the same properties. In fact, Γ(OP,OQ) is a reflexive polytope (with a regular unimodular triangulation) if and only if P and Q have a common linear extension ([7, Corollary 2.2]). Moreover, Γ(OP,CQ) and Γ(CP,CQ) are always reflexive polytopes with regular unimodular triangulations ([10, Corollary 1.2] and [9, Corollary 1.3]). In [9, 29], combinatorial properties of these polytopes, for example, their volumes, are studied.

3. REFLEXIVE POLYTOPES ARISING FROM THE STABLE SET POLYTOPES OF PERFECT GRAPHS In this section, we give a family of reflexive polytopes with regular unimodular triangulations arising from the stable set polytopes of perfect graphs. In particular, we show that Question 0.4 has a positive answer for the stable set polytopes of perfect graphs. First, we recall what perfect graphs are and introduce the stable set polytopes of finite simple graphs. Let G be a finite simple graph on the vertex set [d] and E(G) the set of edges of G. (A finite graph G is called simple if G possesses no loop and no multiple edge.) A subset W [d] is called stable if, for all i and j belonging to W with i = j, ⊂ ∕ one has i, j E(G). We remark that a stable set is often called an independent set. { } ∕∈ A clique of G is a subset W [d] which is a stable set of the complementary graph G ⊂ of G. The clique number ω(G) of G is the maximal cardinality of a clique of G. The chromatic number χ(G) of G is the smallest integer t 1 for which there exist stable ≥ set W ,...,W of G with [d] = W W . In general, it follows that ω(G) χ(G). 1 t 1 ∪ ··· ∪ t ≤ A finite simple graph G is said to be perfect ([2]) if, for any induced subgraph H of G including G itself, one has ω(H) = χ(G). Perfect graphs include many important classes of graphs, for example, chordal graphs and comparability graphs. Moreover, it is known that the complementary graph of a perfect graph is perfect ([2]). This characterization of perfect graphs is called the perfect graph theorem. Recently, a stronger characterization of perfect graphs, which is called the strong perfect graph theorem, is known. An odd hole is an induced odd cycle of length 5 and an odd antihole is the complementary graph of ≥ an odd hole.

Proposition 3.1 ([2, Strong Perfect Graph Theorem]). A finite simple graph G is perfect if and only if G has no odd hole and no odd antihole as induced subgraph. Next, we introduce the stable set polytopes of finite simple graphs. Let S(G) denote the set of stable sets of G. One has 0/ S(G) and i S(G) for each i [d]. The stable set d ∈ { } ∈ ∈ polytope QG R of G is the (0,1)-polytope which is the convex hull of ρ(W) : W ⊂ 7 { ∈ d S(G) in R . Then the dimension QG is equal to d. It is known that every chain polytope } is a stable set polytope. In fact, let P be a finite poset on [d]. Its comparability graph GP is the finite simple graph on [d] such that i, j E(G ) if and only if i < j or j < i in P. { } ∈ P Then a stable set of GP corresponds to an antichain of P. Moreover, one has CP = QGP . Since every comparability graph is perfect, the class of the chain polytopes is contained in the class of the stable set polytopes of perfect graphs. We see a characterization of perfect graphs in terms of the stable set polytopes. Proposition 3.2 ([19, Example 1.3 (c)]). Let G be a finite simple graph on [d]. Then G is perfect if and only if QG is compressed.

Now, we consider the lattice polytope Ω(QG) for a perfect graph G on [d]. In particular, we see that the toric ideal of this lattice polytope is squarefree with respect to some monomial order on K[x]. Let

K[Q] := K[ xS,yS S S(G) z ] { } ∈ ∪ { } denote the polynomial ring over K and deﬁne the surjective ring homomorphism πQ by the following:

πQ : K[Q] K[Ω(QG)] by setting • →ρ(S d+1 ) ρ(T d+1 ) πQ(xS) = t ∪{ } s,πQ(yT ) = t− ∪{ } s and πQ(z) = s where S,T S(G). Then the toric ideal I is the kernel of π . ∈ Ω(QG) Q Next, we introduce monomial order

(resp. in

By combining Lemma 1.1 and Proposition 3.3, we can show that Ω(QG) is a reflexive polytope with a regular unimodular triangulation. Moreover, we can give a polyhedral characterization of perfect graphs. Theorem 3.4 ([13, Theorem 1.1 (b)]). Let G be a finite simple graph on [d]. Then the following arguments are equivalent: (1) G is perfect; (2) Ω(QG) is a reflexive polytope with a regular unimodular triangulation; 8 (3) Ω(QG) has a regular unimodular triangulation. Hence we can obtain the following corollary.

Corollary 3.5. Let G be a perfect graph on [d]. Then one has rdim(QG) = d + 1. Remark 3.6. In [13], a lager family of reﬂexive polytopes is given. In fact, for two ﬁnite simple graphs G1 and G2 on [d], the lattice polytope Ω(QG1 ,QG2 ) is studied. By the same technique, it follows from [13, Theorem 1.1 (b)] that the following arguments are equivalent:

(1) G1 and G2 are perfect;

(2) Ω(QG1 ,QG2 ) is a reﬂexive polytope with a regular unimodular triangulation;

(3) Ω(QG1 ,QG2 ) has a regular unimodular triangulation.

Remark 3.7. In [21], the lattice polytope Γ(QG1 ,QG2 ) for two ﬁnite simple graphs G1,G2 on [d] is studied. The two lattice polytopes Ω(QG1 ,QG2 ) and Γ(QG1 ,QG2 ) have a very similar but different property. In fact, it follows from [21, Theorem 2.8] that the following arguments are equivalent:

(1) G1 and G2 are perfect;

(2) Γ(QG1 ,QG2 ) is a reﬂexive polytope with a regular unimodular triangulation;

(3) Γ(QG1 ,QG2 ) is a reﬂexive polytope.

Remark 3.8. In [12], the lattice polytopes Ω(QG,OP) and Γ(QG,OP) for a finite simple graph G on [d] and a finite poset P on [d] are studied. In fact, it follows from [12, Theorem 1.2] that the following arguments are equivalent: (1) G is perfect; (2) Ω(QG,OP) has a regular unimodular triangulation; (3) Γ(QG,OP) is a reflexive polytope.

In particular, if G is perfect, then each of Ω(QG,OP) and Γ(QG,OP) is a reﬂexive polytope with a regular unimodular triangulation.

4. REFLEXIVE POLYTOPES ARISING FROM EDGE POLYTOPES

When we show that Ω(P) is reflexive for a lattice polytope P Zd of dimension ⊂ d by using Lemma 1.1, we need to assume that P possesses the integer decomposition property. In fact, order polytopes, chain polytopes and the stable set polytopes of perfect graphs are compressed, hence they possess the integer decomposition property. In order to show that Question 0.4 has a positive answer for a class of (0,1)-polytopes which do not necessarily possess the integer decomposition property, we should consider other approaches. In this section, we introduce a linear algebraic technique to show that a lattice polytope which does not necessarily possess the integer decomposition property is reflexive, and we give one family of reflexive polytopes arising from the edge polytopes of finite simple graphs. 9 For two d d integer matrices A,B, we write A ∼ B if B can be obtained from A by × some row and column operations over Z. Let X = x1,...,xn be a set of lattice points in d { } R . Given a subset I = i1,...,id+1 [n], let XI be the (d + 1) (d + 1) matrix whose { } ⊂ × ith row vector is (xi,1).

Lemma 4.1 ([16, Proof of Theorem 2.1]). Let X = x ,...,x be a set of lattice points { 1 n} in Rd and P Rd a lattice polytope of dimension d all of whose vertices belong to X. ⊂ Assume that for any subset I = i ,...,i [n] with det(X ) = 0, it follows that for { 1 d+1} ⊂ I ∕ some integer 0 s d, ≤ ≤ 1 .. s � . 0 � � 1 � XI ∼ � � . � 2 � � � � � . � � .. � � 0 � � 2 � � � � � Then Ω(P) is reﬂexive. � �

In order to give a family of reflexive polytopes, to find X which satisfies the condition of Lemma 4.1 is an interesting problem. In fact, by using this lemma, we can get a large family of reflexive polytopes.

Proposition 4.2 ([16, Proposition 2.3]). Set

d X = 0 ei : 1 i d ei + e j : 1 i j d R . { } ∪ { ≤ ≤ } ∪ { ≤ ≤ ≤ } ⊂ Then for any subset I = i ,...,i [n] with det(X ) = 0, it follows that for some { 1 d+1} ⊂ I ∕ integer 0 s d, ≤ ≤ 1 .. s � . 0 � � 1 � XI ∼ � � . � 2 � � � � � . � � .. � � 0 � � 2 � � � � � By combining Lemma 4.1 an�d Proposition 4.2, we can o�btain the following theorem.

Theorem 4.3 ([16, Theorem 2.1]). Let P Rd be a full-dimensional lattice polytope all ⊂ of whose vertices belong to

0 e : 1 i d e + e : 1 i j d . { } ∪ { i ≤ ≤ } ∪ { i j ≤ ≤ ≤ } Then Ω(P) is reflexive. 10 The class of lattice polytopes which satisfy the condition of Theorem 4.3 contains a well-known large family of (0,1)-polytopes, namely the edge polytopes of finite simple d graphs. Let G be a simple graph on [d]. The edge polytope PG R of G is the convex ⊂ hull of all vectors e + e such that i, j E(G). This means that the edge polytope of i j { } ∈ PG of G is the convex hull of all row vectors of the incidence matrix AG of G, where AG is the matrix in 0,1 E(G) [d] with { } × 1 if v e, ae,v = ∈ �0 otherwise. Moreover, the dimension of P equals rank(A ) 1. Hence edge polytopes are not full- G G − dimensional. However, given an edge polytope PG, one can easily get a full-dimensional unimodularly equivalent copy P of PG by considering the lattice polytope defined as the convex hull of the row vectors of AG with some columns deleted. This implies that every edge polytope PG is unimodularly equivalent to a lattice polytope which satisfies the condition of Theorem 4.3. In particular, Ω(PG) is unimodularly equivalent to Ω(P). Hence we can get the following corollary.

Corollary 4.4. Let G be a finite simple graph on [d]. Then one has rdim(PG) = dim(PG)+ 1. In general, an edge polytope does not possess the integer decomposition property. In fact, it is known when the edge polytope of a connected finite simple graph possesses the integer decomposition property. Proposition 4.5 ([18, Corollarly 2.3]). Let G be a connected finite simple graph on [d]. Then PG possesses the integer decomposition property if and only if for any two odd cycles C and C′ of G having no common vertex, there exists an edge of G joining a vertex of C with a vertex of C′.

However, even if the edge polytope PG of a connected ﬁnite simple graph G possesses the integer decomposition property, the reﬂexive polytope Ω(PG) does not always possess the integer decomposition property. Example 4.6. Let G be the connected simple graph as follows:

G: ❅ � � ❅ � � ❅ � ❅ � ❅ � � ❅ � � � ❅ � ❅ � ❅ � ❅ � � 11 Then G satisﬁes the condition of Proposition 4.5. Hence PG possesses the integer decomposition property. However, Ω(PG) does not possess the integer decomposition property.

Finally, we characterize when Ω(PG) possesses the integer decomposition property for a connected ﬁnite simple graph G.

Theorem 4.7 ([16, Theorem 3.2]). Let G be a connected ﬁnite simple graph on [d]. Then Ω(PG) possesses the integer decomposition property if and only if G does not contain two disjoint odd cycles.

Remark 4.8. We remark that the condition of Theorem 4.7 characterizes some class of edge polytopes. In fact, a connected ﬁnite simple graph G does not contain two disjoint odd cycles if and only if PG is unimodular, that is, all its triangulations are unimodular ([17, Example 3.6 b]).

Remark 4.9. In [20], the lattice polytope Γ(PG) for a connected finite simple graph G is studied. In fact, if G does not contain two disjoint odd cycles, then Γ(PG) is a reflexive polytope which possesses the integer decomposition property. However, in general, Γ(PG) is not always reflexive. For instance, if G is the connected finite simple graph which appears in Example 4.6, then Γ(PG) is not reflexive and does not possess the integer decomposition property.

REFERENCES [1] V. V. Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom., 3:493–535, 1994. [2] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The strong perfect graph theorem. Ann. of Math., 164:51–229, 2006. [3] D. A. Cox, J. B. Little, and H. K. Schenck. Toric varieties, volume 124 of Graduate Studies in Math- ematics. American Mathematical Society, Providence, RI, 2011. [4] C. Haase and H. V. Melinkov. The reflexive dimension of a lattice polytope. Ann. Comb., 10:211–217, 2006. [5] T. Hibi. Gro¨bner Bases: Statistics and Software Systems. Springer, 2013. [6] T. Hibi, N. Li, T. X. Li, L. Mu, and A. Tsuchiya. Order-Chain Polytopes. Ars Math. Contemp., 16:299– 317, 2019. [7] T. Hibi and K. Matsuda. Quadratic Gro¨bner bases of twinned order polytopes. European J. Combin., 54:187–192, 2016. [8] T. Hibi, K. Matsuda, H. Ohsugi, and K. Shibata. Centrally symmetric configurations of order polytopes. J. Algebra, 443:469–478, 2015. [9] T. Hibi, K. Matsuda, and A. Tsuchiya. Gorenstein Fano polytopes arising from order polytopes and chian polytopes. arXiv:1507.03221, 2015. [10] T. Hibi, K. Matsuda, and A. Tsuchiya. Quadratic Gro¨bner bases arising from partially ordered sets. Math. Scand., 121:19–25, 2017. [11] T. Hibi and A. Tsuchiya. Facets and volume of gorenstein fano polytopes. Math. Nachr., 290:2619– 2628, 2017. [12] T. Hibi and A. Tsuchiya. Reflexive polytopes arising from partially ordered sets and perfect graphs. J. Algebraic Combin., 49:69–81, 2019. 12 [13] T. Hibi and A. Tsuchiya. Reflexive polytopes arising from perfect graphs. J. Combin. Theory Ser. A, 157:233–246, 2018. [14] M. Kreuzer and H. Skarke. Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys., 4:1209–1230, 2000. [15] J. C. Lagarias and G.M. Ziegler. Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canad. J. Math., 43:1022–1035, 1991. [16] T. Nagaoka and A. Tsuchiya. Reflexive polytopes arising from edge polytopes. Linear Algebra Appl., 557:438–454, 2018. [17] H. Ohsugi, J. Herzog, and T. Hibi. Combinatorial pure subrings. Osaka J. Math., 37:745–757, 2000. [18] H. Ohsugi and T. Hibi. Normal polytopes arising from finite graphs. J. Algebra, 207:409–426, 1998. [19] H. Ohsugi and T. Hibi. Convex polytopes all of whose reverse lexicographic initial ideals are squarefree. Proc. Amer. Math. Soc., 129:2541–2546, 2001. [20] H. Ohsugi and T. Hibi. Centrally Symmetric Configurations of Integer Matrices. Nagoya Math. J., 216:153–170, 2014. [21] H. Ohsugi and T. Hibi. Reverse lexicographic squarefree initial ideals and Gorenstein Fano polytopes. J. Commut. Alg., 10:171–186, 2018. [22] H. Ohsugi and A. Tsuchiya. Reflexive polytopes arising from bipartite graphs with γ-positivity asso- ciated to interior polynomials. arXiv:1810.12258, 2018. [23] H. Ohsugi and A. Tsuchiya. Enriched order polytopes and enriched Hibi rings. Eur. J. Math., 2020. [24] H. Ohsugi and A. Tsuchiya. Enriched chain polytopes. Israel J. Math., to appear. [25] A. Schrijver. Theory of Linear and Integer Programing. John Wiley & Sons, 1986. [26] R. P. Stanley. Two poset polytopes. Discrete Comput. Geom., 1:9–23, 1986. [27] B. Sturmfels. Gro¨bner Bases and Convex Polytopes, volume 8 of University Lecture Series. American Mathematical Society, Providence, RI, 1996. [28] S. Sullivant. Compressed polytopes and statistical disclosure limitation. Tohoku Math. J., 58:433 – 445, 2006. [29] A. Tsuchiya. Volume, facets and dual polytopes of twinned chain polytopes. Ann. Comb., 22:875–884, 2018.

(Akiyoshi Tsuchiya) GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF TOKYO, 3-8-1 KOMABA, MEGURO-KU, TOKYO 153-8914, JAPAN Email address: [email protected]