Tope Graph of Oriented Matroids

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1 Tope Graph of Oriented Matroids Clara Brioschi, Legi: st December Introduction and Definitions In this report I will first define tope graphs of oriented matroids and their geometric interpretation. Then, I will outline and prove their main properties. Definition 1. A graph G = (V (G), E(G)) is a pair of finite set of vertices V (G) and a set of edges E(G), which are undirected. A graph G is said to be connected if every pair of vertices in the graph are connected, i.e. ther exists a path from one to the other. For a connected graph G we denote: d G (v, w) = min{ V V E(G), V path connecting v and w} diam(g) = max {d G(v, w)} v,w V (G) Definition 2. Two graphs G and G are called isomorphic if there exists a bijection φ : V (G) V (G ) such that {φ(v), φ(w)} E(G ) {v, w} E(G) and φ is called graph isomorphism. In case that G = G φ is called graph automorphism. Definition 3. Let (E, T ) be a pair of a finite set E and T {+,, 0} E such that all sign vectors in T have the same support. The tope graph of (E, T ) is a graph G with exactly T vertices that can be associated as a bijection L : V (G) T such that: {x, y} E(G) D(L(x), L(y)) is a parallel class of T (2) In particular, if T is the tope graph of an oriented matroid M, then G is called the tope graph of M and L is called its associating bijection. Definition 4 (Alternative Definition of tope graph of an oriented matroid). Let (E, F ) be an oriented matroid with tope set T. Its tope graph G is a graph with exactly f d = T vertices that can be associated by a bijection L : V (G) T such that {x, y} E(G) if and only if L(x) and L(y) have a lower common neighbor in the big face lattice. Remark 1. We can easily prove that Definition 3 and Definition 4 are equivalent by recalling that the parallel classes of T are the same as those of F and by observing that if X, Y T then there exists a (d-1)-face Z F such that Z X and Z Y Z is of the form Z \ D := X \ D = Y \ D and Z D = 0 (1) 1

2 Figure 1: Pseudosphere arrangement and tope graph of an oriented matroid 1.1 Geometric Interpretation of Tope Graphs Consider the pseudosphere arrangement of a d-dimensional oriented matroid (Figure 1 ). We define regions the cells of maximal dimension d-1 and topes their corresponding covectors. Two regions are called adjacent if they have a common (d-2)-dimensional face. This is the case if and only if their corresponding topes X and Y disagree in exactly one element. This defines an adjacency notion for topes. In the tope graph, every vertex corresponds to a tope and two vertices are connected if their corresponding topes disagree in exactly one element, i.e. their corresponding regions in the pseudosphere arrangement are adjacent. (Figure 1 ). Remark 2. In the pseudosphere arrangement a region is adjacent to exactly other three regions. This property is also reflected to the tope grpah, where each vertex is adjacent to just other three vertices. 1.2 Operations between Sign Vectors Let F {, +, 0} E. Definition 5. A Relabeling of F is a set F {, +, 0} E that there exists a bijection φ : F F and a bijection with E finite such such that ψ : {[e] E e is not a loop of F} {[e ] E e is not a loop of F } X e = φ(x) e and ψ([e]) = [e ] for every e E and e E no loops of E and E respectively. A Reorientation of F is a set of sign vectors of the form: { S X X F } for some S E F is said to be isomorphic to F if F can be obtained from F by relabeling first and reorienting. 2

3 Relabeling, reorientation and isomorphism define equivalence relations for sets of sign vectors. Relabeling and so isomorphism motivate the introduction of operation of deleting parallel elements and loops. In this way we can obtain an isomorphic set of sign vectors without loops and whose parallel classes contain only one element. Definition 6. F is called simple if it contains no loops and no parallel elements e f. An oriented matroid (E, F ) is called simple if F is simple. A simplification of F is a set of sign vectors which is isomorphic to F. The followiong lemma shows that we can consider tope graphs of simple oriented matroids without loss of generality. Lemma 1. The tope graph of a set T is equal to the tope graph of any simplification of T. More generally, the tope graphs of any isomorphic sets of sign vectors are equal. Proof. The proof is very simple. You just need to remember that we have defined tope graphs by relating to the concept of big face lattice (Definition 4 ), which we know to be invariant under the operations of relabeling and reorientation. 2 Properties of Tope Graphs There are two main problems in which we are concerned reguarding tope graphs: Characterization Problem: given a tope graph G = (V (G), E(G)), decide whether G is the tope graph of some oriented matroid; Reconstruction Problem: given a tope graph G = (V (G), E(G)) of some oriented matroid, find an oriented matroid M such that G is the tope graph of M. In order to deal with these problemes, we need to discuss some important properties of tope graphs. In particular, we first introduce the notions of L 1 systems and acycloids which generalize the tope sets of oriented matroids. Definition 7. Let (E, T ) be a pair of a finite set E and a set T {, +} E. We define the following properties: (A1): X, Y T such that X Y, e D(X, Y ) and Z T, such that: Z e = X e and Z \ e = X \ e i.e., if Z and X differs only by the element e E. (A2): (symmetry) if X T then X T. Definition 8. A pair (E, T ) of a finite set E and a set T {, +} E is called L 1 system if it satisfies property (A1). If it also satisfies property (A2), then it is called an acycloid. The following lemma proves that the acycloid is a generalization of an oriented matroid. 3

4 Lemma 2. Let M = (E, F ) be a simple oriented matroid. Then (E, T ) is an acycloid, where T is the tope set of M. In order to prove this lemma, we need the following property of tope sets. Recall 1 (Orientation Property of tope sets of an oriented matroid). For every X and Y in T, with X different from Y, there exists a parallel class S contained in D(X, Y ), such that S X is in T. Proof. Since M is a simple oriented matroid it has no loops, thus its tope set T {, +} E. We now prove that property (A1) is equivalent to the orientation property in Recall 1. We know that F is simple, thus every parallel class has exaclty one element. This means that the set S in the orientation property is composed of only one element, say e. Thus, this property becomes: X, Y T, X Y, e D(X, Y ) such that e X T Z T such that Z e = X e and Z \ e = X \ e which is exactly property (A1). Property (A2) follows by the symmetry of covectors. Property 1. If (E, T ) is an L 1 system and G its tope graph with associating bijection L : V (G) T, then Proof. By definition of tope graph : E(G) = {{x, y} D(L(x), L(y)) = 1} (3) {x, y} E(G) D(L(x), L(y)) is a parallel class of T, but an L 1 system is simple, thus its parallel classes contain only one element. It follows that in this case we can write Equation 2 as: {x, y} E(G) D(L(x), L(y)) = {e} D(L(x), L(y)) = 1. The following property states that tope graphs of an L 1 system can be isometrically embedded in some higher-dimensional hypercube. In particular, isometrically means that the distances in the tope graph are the same as in the hypercube. Property 2. If (E, T ) is an L 1 system and G its tope graph with associating bijection L : V (G) T, then d G (x, y) = D(L(x), L(y)) x, y V (G) (4) 4

5 Proof. For semplicity, let us denote X := L(x) and Y := L(y). We prove this property by induction on the dimension of the separating set D(X, Y ). If D(X, Y ) = 0 then x = y by Property 1 and d G (x, y) = 0. If D(X, Y ) = 1, then the result follows by Property 1. Now, suppose D(X, Y ) > 1. We first prove that d G (x, y) D(L(x), L(y)). By assumption, X Y, thus by property (A1) there exists an element e in D(X, Y ) and Z T such that Z e = X e and Z \ e = X \ e. It follows that there exists z V (G) such that Z = L(z). Obviously, D(X, Z) = 1 and D(Z, Y ) = D(X, Y ) 1. Thus, d G (x, z) = 1, which means that {x, z} E(G). By induction, d G (z, y) = D(X, Y ) 1. It follows that d G (x, y) d G (x, z) + d G (z, y) = D(X, Y ) Now, the other direction is directly implied by Property 1 : D(L(x), L(y)). d G (x, y) Property 3. If (E, T ) is an acycloid and G its tope graph with associating bijection L : V (G) T, then for every vertex v in V (G) there exists only one vertex v in V (G) such that In particular, v is called antipode of v in G. d G (v, v) = diam(g) (5) Proof. Consider v V (G). By property (A2) of an acycloid, L(v) T. Let v be the vertex in V (G) such that L(v) = L( v). By Property 2 d G (v, v) = D(L(v), L( v)) = E which is the maximal distance between any vertices in V (G). Uniqueness follows by considering that the maximal distance is attanined only between two vertices whose separating set contain all the elements in E, and by remembering that the acycloid is simple. Now, we define a very useful object for the analysis of tope graphs. Definition 9. Let (E, T ) be an L 1 system and G its tope graph with associating bijection L : V (G) T. For e E we define the edge class of e as follows: E e := {{v, w} E(G) D(L(x), L(y)) = {e}} For an edge {v, w} E(G) we define: C(v, w) := {x V (G) d G (x, v) < d G (x, w)} We can show that edge classes are defined by the graph G itself, independently of L. Property 4. Let G be the tope graph of an L 1 system and L : V (G) T its associating bijection. Let {v, w} E(G) be an arbitrary edge in G, say {v, w} E e for some e E. Then: and C(v, w) = {x V (G) L(x)e = L(v) e } E e = {{v, w } E(G) v C(v, w) and w C(w, v)}. 5

6 Proof. For semplicity, let us denote V := L(v), W := L(w) and X := L(x), for an x V (G). By Property 2 d G (x, v) = D(X, V ) and d G (x, w) = D(X, W ). We know that x C(v, w) d G (x, v) < d G (x, w) D(X, V ) < D(X, W ) but this is true because {v, w} E e by hypothesis, i.e X e = V e = W e. Thus x C(v, w) if and only if X e = V e. We have proved the first statement. Now we need to prove that E e can be written as above. Let V := L(v ), W := L(w ) for some {v, w } in E(G). v C(v, w) V e = V e w C(w, v) W e = W e Thus, D(V, W ) = {e} implies that e D(V, W ), i.e {v, w } E e. On the other hand, after possibly intechanging V and W, which does not change the edge size since {v, w } = {w, v }, if e D(V, W ) then v C(v, w) and w C(w, v) which proves the second statement. Property 5 (Tope Graph of Oriented Matroids of Rank 2). The tope graph of an oriented matroid M = (E, F ) of rank 2 is a cycle of even length 2n, where n is the number of parallel classes in F/E 0, where E 0 is the set of loops. In order to prove this property, we need to recall the diamond property: Recall 2 (Diamond Property). Let M = (E, F ) be an oriented matroid and X, Y ˆF (big face lattice) such that X Y and rank M (Y ) rank M (X) = 2. Then there exists exactly two covectors Z 1, Z 2 F with the property that X Z i Y for i {1, 2}. Proof. Let M be an oriented matroid with rank(m) = 2, M = (E, F ), and let L : V (G) T be its associating bijection. Because rank(m) = 2 we know that 0 / T, (0 T means that 0 is an independent se but it is not). By symmetry of covectors: X T = X T thus, T = 2n for some integer n > 0. We also know now that the edges of G correspond to the cocircuits of the oriented matroid. By this and by the diamond property we can infer that the degree of every vertex is 2. This implies that G consists of a set of cycles and by property 2 G is connected. Thus, G has the form of a cycle of lenght 2n where n = diam(g) = E where E si the ground set of any simplification of M. By definition, E is equal to the number of parallel classes of non-loop elements. Remark 3. In tope graph of oriented matroids of rank 2 every edge class contains two edges which have opposite positions in the cycle. (Fig. 1) 6

7 Figure 2: Tope graph of an oriented matroid of rank 2 3 Property 5: Separability of Uncut Topes Definition 10. Let M = (E, F ) be a simple oriented matroid with tope set T, let f E arbitrary. Define: T := {Z T Z f = and f Z / T } T + := {Z T Zf = + and f Z / T } T 0 := {Z T f Z T } The sets T and T + are set of topes not cut or uncut by f. The motivation for this name comes from considering sphere arrangement and the deletion minor M \ f. If we insert a new sphere S f in the arrangement according to M \ f, then some of the regions of the minor remains unchanged and some are cut by S f into two different regions. The topes T corresponds to the uncut regions on the side of S f, T + to the uncut regions on the + side of S f and T 0 to the regions obtained by a cut. Theorem 1. Let M = (E, F ) be a simple oriented matroid with tope set T. Let f be an arbitrary element of E. Then, for every two topes X, Y T there exists a sequence X = Z 0,... Z k = Y such that Z i T for i {0,... k} and D(Z i 1, Z i ) = 1 for i {0,... k}. This theorem proves that the vertices in T are connected in the sense of adjacency in tope graphs. Of course, by symmetry it also holds for topes in T +. Before proving it, we state its main consequence: Remark 4. In order to see that Theorem 1 hols, the line arrangement has to be embedded on the front side of a sphere with a corresponding extension to the back side. Property 6. Let G be the tope graph of an oriented matroid M = (E, F ) and E f E(G) an edge class. Denote by V 0 the set of vertices incident to some edge in E f, then the subgraph of G induced by the vertices V (G) \ V 0 has either no or exactly two connected components. 7

8 Proof. Property 6 is an immediate consequence of Theorem 1. Let L : V (G) T {, +} E be the associating bijection of G. By definition of edge class there exists f E such that: L(V 0 ) = T 0 If T \ T 0 is the empty set, then there is nothing to prove. Thus, let T \ T 0. By T heorem1 this implies that there are exactly two connected components in the subgraph of G induced by the vertices V (G) \ V 0, one corresponding to T and one to T +. Definition 11 (Separable Tope Graph). Let G be the tope graph of an L 1 system and E f an edge calss in G. We say that G is separable w.r.t. E f if the separability holds for E f : the subgraph of G induced by the vertices V (G) \ V 0 has either no or exactly two connected components, where V 0 denotes the set of vertices incident to some edge in E f. We call G separable if G is separable w.r.t. all edge classes. Now, we can prove Theorem 1, but we need first to recall the definition of Oriented Matroid Program and some property of contraction minors. Recall 3 (Oriented Matroid Program). Let M = (E, F ) be an oriented matroid and f, g E two different elements. Let X, Z be sign vectors on E. The oriented matroid program OMP (M, g, f) is the problem to find an optimal sign vector X, which means finding a sign vector X which is feasible (i.e. X F and X \ f 0 and X g = +) and there is no augmenting direction for X. In particular, a sign vector Z is called an augmenting direction for X if Z is a direction (i.e. Z F and Z g = 0), Z \ f 0 and Z f = +. Recall 4. Let M = (E, F ) be an oriented matroid, E E, X F. Then: { rank(m) if e is a loop rank(m/e) = rank(m) rank M (e) = rank(m) 1 otherwise Recall 5 (Fundamental Theorem of OMP). Every oriented matroid program (M, g, f) is exactly one of optimal, unbounded or infeasible. Proof of Theorem 1. We prove the theorem just for T. The proof for T + is analogous by symmetry. The proof is by induction on the rank of the matroid M. Let r = rank(m). r < 2 then of course the proof is true; r = 2 then the result follow by Property 5 ; r 3. If T =, then there is nothing to prove. So assume T. Let X, Y T. Since X f = Y f = we have that X Y. Moreover, by the reorientation property (A1) in Definition 7 we know that: g D(X, Y ) = E \ D(X, Y ) s.t. g X T X T implies g f. By how we defined X and Y we know that X g = Y g 0. Without loss of generality we can assume X g = Y g = +. We need to consider now two different cases: either g Y T or g Y / T. 8

9 1. Let g Y T. Consider the contraction minor M/g, i.e. the contraction of M to faces which contain g in the zero support. M/g is an oriented mtroid, not necessarily simple, whose rank is rank(m) 1 by a property of contraction minors (see Recall 4 ). Let M be a simplification of M/g where the parallel class containing f is represented by f. Note that by definition, X \ g and Y \ g M/g and denote with X and Ỹ their images in M. Then X, Ỹ T where T is defined for M as T was defined for M. By inductive hypothesis there exists a sequence X = Ũ 0,..., Ũ k = Ỹ in T s.t. D(Ũ i 1, Ũ i ) = 1 for i {1,..., k} Consider now i {0,..., k}. Ũ i T implies that U i T s.t. U i g = + and g U i = T where Ũ i is the image of U i \ g in M. Moreover, Uf i =, and at most one between f U i and {f,g} U i is in T, it implies that at least one between U i and g U i is in T. We define { Û i U i if U := i T g U i otherwise, i.e if U i T Because Û 0 = X and Û k = Y, it only remains to show that Û i 1 and Û i are connected within T for all i {1,..., k} in the sense of the theorem. Consider now i {0,..., k}. By Property 2, there exists two sequences U i 1 = V 0,..., V d = U i and g U i 1 = W 0,..., W d = g U i with d = D(V j 1, V j ). Now we need again to distinguish two cases: If at least on of the two sequences lies entirely in T for i {0,..., k}, then the result follows (you just need to combine all these sequences in T ); Assume that for some i {0,..., k} none of the two sequences in entirely in T. This means that s, t {0,..., d} s.t. V := f V s T and W := f W t T. If we now apply covector elimination property to V, W and g we obtain Z F such that Z g = 0 and Z e = (V W ) e for e / D(V, W ), i.e. Z e = V e = W e for E / D(V, W ). In particular, Z f = +. 9

10 Note that D := D(U i 1, U i ) is a parallel class of M/g, thus Z D = 0, Z D = Û i 1 D, or Z D = Û D i. With also D(V, W ) D {g} it follows that Z Û i 1 = f Û i 1 T or Z Û i = f Û i T, which is a contradiction. 2. Let now g Y / T. We show that Y is connected to some Y in T in the sense of the theorem, then the result follows by part 1.. Because we know that reorientation does not affect connectedness within T by Lemma 1, without loss of generality we can assume Y e = + for all e E \ {f}. Consider now the oriented matroid program (M, g, f) (as in Recall 3 ). No unbounded augmenting direction Z F exists, otherwise we would have that Z Y = f Y T which is a contradiction. Because of this and also because Y is feasible for (M, g, f), by Recall 5 there exists an optimal solution U F for (M, g, f). Note that U \ f 0, U g = +, U f 0 because U f = + implies that U Y = f Y T. Let now: V := U Y T By Property 2, there exists a sequence Y = W 0,..., W d = V T s.t. D(W i 1, W i ) = 1 for i {1,..., d} where d = D(Y, V ). Because Y g = + and V g = U g =, there exists k {1,..., d} such that { Wg i + ifi < k = ifi = k Set now Y := W k 1, then g Y = W k T. It only remains to show that W i T for i {1,..., k 1}. Assume by contradiction that W i / T for some i {1,..., k 1}. This means that W T s.t. W \ f = W i \ f and W f = +. Now, apply the covector elimination to W, U and g. It results into the existence of Z F such that Z g = 0 and Z e = (W U) e for E / D(W, U). In particular, we obtain also that Z f = + and for all E f such that U e = 0, we have V e = Y e = +. So, also W e = + and Z e = W e = +. This means that Z is an augmenting direction for U, in contradiction to the optimatility of U. 10