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A new lower bound for the positive semidefinite minimum rank of a graph
Linear Algebra and its Applications 438 (2013) 1095–1112
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Linear Algebra and its Applications journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / l a a
A new lower bound for the positive semidefinite minimum rank of a graph Andrew M. Zimmer Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States
ARTICLE INFO
ABSTRACT
Article history: Received 26 June 2011 Accepted 9 September 2012 Available online 6 October 2012
The real positive semidefinite minimum rank of a graph is the minimum rank among all real positive semidefinite matrices that are naturally associated via their zero-nonzero pattern to the given graph. In this paper, we use orthogonal vertex removal and sign patterns to improve the lower bound for the real positive semidefinite minimum rank determined by the OS-number and the positive semidefinite zero forcing number. © 2012 Elsevier Inc. All rights reserved.
Submitted by S. Fallat AMS classification: 05C50 15A03 15A18 15B35 15B57 Keywords: Positive semidefinite minimum rank Minimum rank Sign patterns Zero forcing number Positive semidefinite zero forcing number Ordered set number
1. Introduction A graph G = (V , E ) consists of a set of vertices V and a set of edges E, where the elements of E are unordered pairs of vertices. The order of G, denoted |G|, is the cardinality of V . A graph is simple if it has no multiple edges or loops. The entries of an n-by-n Hermitian matrix A = (aij ) over the complex numbers C naturally correspond to a simple undirected graph G(A) with vertex set {1, . . . , n} and edge set {{i, j} : aij = 0, 1 i < j n}. Observe that the diagonal entries of A have no impact on the structure of G(A). E-mail address: [email protected] 0024-3795/$ - see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2012.09.001
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Within this correspondence a number of extremal problems arise, one of recent interest is the minimum rank problem, which seeks to determine the smallest possible rank of any real symmetric matrix associated to a given graph (see the 2007 survey by Fallat and Hogben [7]). Define S (G)
= {A ∈ Mn (R) : AT = A, G(A) = G}.
The minimum rank of G is defined to be mr(G)
= {rank(A) : A ∈ S (G)}.
In this paper, we consider the related problem of determining the minimum rank among positive semidefinite (psd) matrices associated to a given graph (see the 2011 survey by Fallat and Hogben [8]). For K ∈ {R, C} define PK (G)
= {A ∈ Mn (K) : A psd, G(A) = G}.
The real or complex positive semidefinite minimum rank of G is defined to be K mr+ (G)
= {rank(A) : A ∈ PK (G)}.
The real and complex parameters are different, for instance Barioli et al. [1] constructed a graph G for C (G) < mrR (G). which mr+ + A major problem when considering these rank parameters is developing combinatorial conditions K (G) or mr(G) from above or below. Along these lines, the on the graph G = (V , E ) that bound mr+ notion of a zero forcing set was introduced in [9] to bound mr(G) from below. A set Z ⊂ V is a zero forcing set if it satisfies a combinatorial condition involving vertex coloring. One then defines the zero forcing number, denoted Z (G), to be the cardinality of the smallest zero forcing set. By studying the support of vectors in the null space of any matrix representation of G one can show that |G| − Z(G) is a lower bound for mr(G). Theorem 1.1 [9, Proposition 2.4]. |G| − Z(G)
mr(G).
K (G). Hackney et al. [10] introduced the notion of an OS-set to construct a lower bound for mr+ A set S ⊂ V is a OS-set if it satisfies a combinatorial condition involving connectivity in various induced subgraphs. The OS-number, OS(G), is the cardinality of the largest OS-set. By considering K (G). vector representations of a graph G one can show that OS(G) is a lower bound for mr+
Theorem 1.2 [10, Proposition 3.3]. OS(G)
K (G). mr+
Although the two lower bounds were developed independently, it turns out that they are almost “dual” to one another. Mitchell et al. [14, Theorem 2.10] showed that if Z is a zero forcing set then V \ Z is an OS-set. The reverse is not always true: if S is an OS-set we have no guarantee that V \ S is a zero forcing set. However Mitchell et al. defined an OS-number like parameter which equals |G| − Z(G). Shortly after Barioli et al. [1] defined a zero forcing number like parameter called the positive semidefinite zero forcing number, Z+ (G), such that
|G| − Z+ (G) = OS(G). C (G) Hackney et al. [10] showed that mr+
= |G| − Z+ (G) for all chordal graphs and all graphs on six C (G) for all graphs. or fewer vertices. Based on this evidence they conjectured that |G| − Z+ (G) = mr+ But Mitchell et al. [14] constructed examples for which K (G) − OS(G) mr+
K = mr+ (G) − (|G| − Z+ (G))
can be arbitrarily large. In this paper we specialize to the real case and introduce a new lower bound R (G) called the signed reduction number, denoted by R (G). We obtain the following relationship for mr+ s with the previously defined parameters.
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Theorem 1.3. |G| − Z+ (G)
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R (G). = OS(G) R s (G) mr+
We further show that |G| − Z+ (G) = OS(G) In particular we prove the following theorem.
R (G) for the graphs considered in [14]. < R s (G) = mr+
R (G)} and E = {G simple graph : R (G) = = {G simple graph : |G| − Z+ (G) = mr+ s s
Theorem 1.4. Let E
R (G)}. Then mr+
(1) E ⊂ Es , (2) ML8 ∈ Es \ E, (3) both E and Es are closed under vertex sums and graph joins. Here ML8 is the Möbius ladder on 8 vertices and the operations of vertex sums and graph joins are / E and E is closed under vertex sums and graph joins follows given in Section 6. The fact that ML8 ∈ from the work in [14]. The motivation for the word “reduction” comes from a new interpretation of the OS-number developed by Mitchell et al. [15] and described in Section 3 of this paper. The motivation for the word “signed” follows from the use of sign patterns to construct R s (G). Given a simple graph G = (V , E ), a sign pattern of G is a function f : E → {1, −1}. We call the pair (G, f ) a signed graph. If (G, f ) is a signed graph then PR (G, f ) ⊂ PR (G) denotes the set of all real psd matrices A = (aij ) such that:
• aij < 0 if ij ∈ E and f (ij) = −1, • aij > 0 if ij ∈ E and f (ij) = 1, • aij = 0 if ij ∈ / E. The positive semidefinite minimum rank of a signed graph (G, f ) is defined to be R mr+ (G, f )
= min{rank A : A ∈ PR (G, f )}.
In Section 4, we construct a lower bound for signed graphs, denoted by R (G, f ), such that R |G| − Z+ (G) R (G, f ) mr+ (G, f ).
(1)
By exploiting the fact that R mr+ (G)
R = min mr+ (G, f ) f
(2)
R (G), namely one obtains a new lower bound for mr+
R s (G)
= min R (G, f ). f
By the inequality in 1, we then have R |G| − Z+ (G) R s (G) mr+ (G).
1.1. Some history The basic problem of understanding how the zero-nonzero pattern or sign pattern of a matrix A determines its spectral properties is an old and well studied problem. In this short subsection, we will mention some previous work involving sign patterns and minimum rank. A more in depth introduction can be found in [8, Section 5.3] or [11, Chapter 33]. Given a signed simple graph (G, f ) one can naturally define S (G, f ) ⊂ S (G) and the minimum rank parameter mr(G, f ). With slightly different notation, mr(G, f ) has been computed when G is a tree [5,12]. Several authors have also considered a non-symmetric variant of this parameter: given a
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m-by-n sign pattern Z, what is the minimum rank of a matrix A with sign pattern Z? In this context, a number of authors have explored interesting connections between the minimum rank of full sign patterns and communication complexity, for additional information see the book by Lokam [13]. R (G, f ) will behave very differently than the corresponding minimum We suspect the parameter mr+ rank parameter mr(G, f ). One key difference is that the set S (G) is closed under additive inverses: if A ∈ S (G) then −A ∈ S (G). This “closure” does not hold for the set PR (G). As a result of this difference, the main results of Section 5 of this paper have no minimum rank analogue. For instance if Km is the complete graph on m-vertices and n : E (Km ) → {+1, −1} is the all negative sign pattern then R (K , n) = m − 1 (see Theorem 6.3). mr(Km , n) = 1 but mr+ m In the language of vector representations, Parson and Pisanski [16] and Šinajová ˇ [17] considered R (G, n) when n is the all negative sign pattern: n(e) = −1 for all e ∈ G. the problem of computing mr+
R (G, n) = |G| − 1 when G is connected. In Section 5 we will In this case, Šinajová ˇ [17] proved that mr+ prove this result using the parameter R (G, n). The central idea of this paper, which does not appear to be present in any of the papers cited above, is that sign patterns can be exploited to obtain better bounds on the minimum rank of a set of matrices determined by a fixed zero-nonzero pattern.
1.2. Outline of paper Section 2 is devoted to some preliminaries. In Section 3 we describe the reduction number. Section 4 is the beginning of our new results, there we refine the reduction number for sign patterns. In Section 5 we describe a natural equivalence relation on the set of sign patterns of a fixed graph. In Section 6 we R (G, f ) for some special cases including when G is a cycle or tree. In Section 7 we prove compute mr+ Theorem 1.4. Finally Section 8 is devoted to proving a lemma used in Section 7. 2. Preliminaries 2.1. Graph theory As our only concern is undirected graphs, an edge {v, u} will sometimes be written as vu. The subgraph G[R] of G induced by R ⊂ V (G) is the subgraph of G with vertex set R and edge set consisting of those edges of G where both vertices are elements of R. For a vertex w of a graph G, let N (w) denote the set of all vertices adjacent to w in G, called the neighborhood of w in G. By the closed neighborhood of w, denoted N [w], we mean {w} ∪ N (w). Let Kn denote the complete graph on n vertices, Cn the cycle on n vertices, and Pn the path on n vertices. If H and K are graphs, let H K denote the disjoint union of H and K. If G is a graph let nG
= G · · · G . n copies
2.2. Vector representations
Given a list of n column vectors in Kd , X
= (x1 , . . . , xn ), let X be the matrix [x1 . . . xn ]. Then X ∗ X
with regard to the Euclidean inner product. Its associated is a psd matrix called the Gram matrix of X x1 , . . . , xn , and edges corresponding to simple graph G has n vertices corresponding to the vectors is a vector nonzero inner products among those vectors. Since X ∗ X ∈ PK (G) for the graph G, we say X , we mean the dimension of the span of the vectors in X , which is equal to representation of G. By rank X rank X ∗ X. As every positive semidefinite matrix is a Gram matrix and conversely every Gram matrix K (G) d if and only if there is a vector representation of G in Kd . is positive semidefinite, mr+ This type of vector representation is a particular instance of the more general vector representations studied by Parson and Pisanski [16].
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2.3. Multigraphs A graph G is a multigraph if G is an undirected graph that has no loops but may have multiple edges between vertices. If G is a multigraph then PK (G) denotes the set of all psd matrices with entries in K where:
• aij = 0 if vertex i and vertex j are connected by exactly one edge in G, • aij = 0 if vertex i and vertex j are not adjacent and i = j. No restriction is placed on aij if vertex i and vertex j are connected by more than one edge. The (real or complex) positive semidefinite minimum rank of a multigraph G is defined as: K mr+ (G)
= min{rank A : A ∈ PK (G)}.
Notice that vector representations generalize to multigraphs in an obvious way. 2.4. Orthogonal vertex removal We next present the concept of orthogonal vertex removal. This was introduced by Booth et al. [4], but we will follow the slightly different but equivalent formulation in Mitchell et al. [15]. Consider ⊂ Kd of a graph G = (V , E ). For v ∈ V with vector representative v a vector representation V consider the subspace U = Span{ v} and the orthogonal projection Pv : Kd → U ⊥ onto its orthogonal form a vector representation of a new simple graph G on n − 1 vertices complement. The vectors Pv V element). v=0 (we ignore the Pv To understand the graph G consider the inner product:
Pv u, Pv w = u, w −
u, v v, w v, v
for some u, w ∈ V and their corresponding vector representatives u , w. The expression above shows that G could be one of several possibilities. To model this we use multigraphs. If v ∈ V (G), the orthogonal removal of v from G, denoted G v, is the multigraph modified from the induced subgraph on the vertices V (G) − {v} such that any u, w ∈ N (v) have P additional edges added between them, where P is the product of the number of edges from v to u and from v to w. is always a vector representation of G v when v = 0. If, in addition, V is a By construction Pv V vector representation of minimal rank we obtain the following inequalities: K mr+ (G v) + 1
K rank Pv V + 1 = rank V = mr+ (G).
We record this as a proposition. Proposition 2.1 [4]. Let G to v by a single edge. Then K mr+ (G v) + 1
= (V , E ) be a multigraph with v ∈ V such that there exists w ∈ N (v) adjacent
K mr+ (G).
The existence of such a w is necessary, otherwise there exists minimal vector representations of G in which v is represented by the zero vector. We distinguish such vertices with the following definition. Definition 2.2. Let G = (V , E ) be a multigraph. A vertex v exists w ∈ N (v) adjacent to v by a single edge.
∈ V is called nondegenerate in G if there
Observation 2.3. Let G = (V , E ) be a multigraph. A vertex v ∈ V is nondegenerate in G if and only if v is represented by a nonzero vector in every vector representation of G.
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3. The reduction number In this section we describe an interpretation of the OS-number which was introduced in [15]. Observe that Proposition 2.1 implies a lower bound for the semidefinite minimum rank: for example if v1 is nondegenerate in G then K (G) mr+
K mr+ (G v1 ) + 1 1.
If in addition there is a vertex v2 which is nondegenerate in G v1 then we also have K mr+ (G)
K mr+ (G v1 v2 ) + 2 2.
In general, if there exists a list of vertices (v1 , . . . , vd ) in G such that v1 is nondegenerate in G and vi+1 K (G) d. is nondegenerate in G v1 · · · vi then Proposition 2.1 implies that mr+ With this motivation, the reduction number of a graph G, denoted R(G), is defined to be the following: R (G)
= max{R (G v) + 1 : v ∈ V (G), v nondegenerate in G}
where max ∅
= 0. The discussion above implies the following.
Theorem 3.1 [15, Theorem 4.1]. R (G)
K (G). mr+
If (v1 , . . . , vd ) is a list of vertices of G such that v1 is nondegenerate in G and vi+1 is nondegenerate in G v1 · · · vi , then we say that (v1 , . . . , vd ) is a reduction order of G. Clearly R (G) d if and only if there is a reduction order of length d. Mitchell et al. also show that the reduction number yields the same lower bound as the OS-number and positive semidefinite zero forcing number. Theorem 3.2 [15, Theorem 4.4]. Let G be a graph, then
|G| − Z+ (G) = OS(G) = R (G). Moreover every OS-set is a reduction order, while every reduction order can be permuted to yield an OS-set. Remark 3.3. Notice that the reduction number is defined for multigraphs. Originally the positive semidefinite zero forcing number and OS-number were only defined for simple graphs. However, Ekstrand et al. [6] extended the definition of Z+ (G) to multigraphs and Mitchell et al. [15] extended the definition of OS(G) to multigraphs. 4. Refining the reduction number for sign patterns To motivate the construction of R (G, f ) and R s (G) as described in the introduction, we consider the following example. Example 4.1. Suppose G is a connected simple graph with vertices u, v, w such that G[u, v, w] = K3 . be a vector representation of G and suppose u, v, w represent u, v, w. As in Section 2, let U = Let V Span{ v} and Pv : Rm → U ⊥ be the orthogonal projection onto the orthogonal complement of U. Then
Pv u, Pv w = u, w −
u, v v, w v, v
(3)
could be positive, zero, or negative. This uncertainty is modeled by the vertices u and w being adjacent is actually a vector representation of the signed graph by multiple edges in G v. Now suppose V (G, f ) and f ({u, v}) = f ({u, w}) = f ({v, w}) = −1. Then we know that the expression in Equation 3 is actually negative. The moral of Example 4.1 is that knowing the sign of each inner product in a vector representation can help prevent uncertainty in orthogonal removal. We now define sign patterns for multigraphs,
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construct a signed version of orthogonal vertex removal, and finally define a signed version of the reduction number. 4.1. Sign patterns of multigraphs If (G, f ) is a signed multigraph let PR (G, f ) denote the set of all psd matrices with entries in where:
R
• aij > 0 if every edge e connecting vertex i and vertex j has f (e) = 1, • aij < 0 if every edge e connecting vertex i and vertex j has f (e) = −1, • aij = 0 if vertex i and vertex j are not adjacent and i = j. No restriction is placed on aij if vertex i and vertex j are connected by edges e and e with f (e) The positive semidefinite minimum rank of a signed multigraph (G, f ) is defined as: K mr+ (G, f )
= f (e ).
= min{rank A : A ∈ PR (G, f )}.
Suppose that G is a multigraph, then the definitions above imply that PR (G)
= ∪ PR (G, f )
and so R mr+ (G, f )
R = min mr+ (G, f ). f
Remark 4.2. Let (G, f ) be a signed multigraph and suppose i, j ∈ V (G). Let e1 , . . . , em be the edges incident to i and j. The definition of PR (G, f ) has the following motivation. For A ∈ PR (G, f ) we think of the entry aij as being the sum of m numbers: aij
=
m
bk
k=1
where bk has the same sign as f (ek ). In particular, if each f (ek ) is negative then aij is negative but if some of the f (ek ) are positive and some are negative then aij could be positive, zero, or negative. 4.2. Signed orthogonal vertex removal We are now ready to construct the lower bound R (G, f ) for signed graphs (G, f ). Let G = (V , E ) be a graph with sign pattern f : E → {1, −1}. Let v ∈ V be a vertex, we wish to construct a sign pattern f v for the graph G v. The signed orthogonal removal of v from (G, f ) is the signed graph (G v, f v) where G v is the orthogonal removal of v from G described in Section 2 and f v is defined as follows: let N (v) = {w1 , . . . , wn } and let e1i , . . . , eni i be the edges connecting v to wi in G, then wi and wj have ni nj additional edges between them in G v. Label these edges ei,j,α,β with 1 α ni and 1 β nj . Finally define (f v)(e) = f (e) if e ∈ E (G) and j
(f v)(ei,j,α,β ) = −f (eαi )f (eβ ).
(4)
⊂ Rd is a vector representation of the signed graph (G, f ), we again construct the vector Now if V space U = Span{ v} and the orthogonal projection Pv : Rd → U ⊥ onto its orthogonal complement. is a vector representation of (G v, f v) when v = 0. If, in v = 0 element, Pv V Ignoring the Pv is a vector representation of minimal rank we obtain the following inequalities addition, V R mr+ (G v, f
R v) + 1 rank V = mr+ (G, f ).
We record this as a proposition.
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Proposition 4.3. Let (G, f ) be a signed graph and v v to w have the same sign. Then R mr+ (G v, f
∈ V (G) with w ∈ N (v) such that all edges connecting
R v) + 1 mr+ (G, f ).
The existence of such a w is necessary, otherwise there exists vector representations of (G, f ) in which v is represented by the zero vector. As before, we distinguish such vertices with the following definition. Definition 4.4. Let (G, f ) be a signed graph. A vertex v ∈ V is called nondegenerate in (G, f ) if there exists w ∈ N (v) such that every edge connecting v and w has the same sign. Observation 4.5. Let (G, f ) be a signed graph. A vertex v ∈ V is nondegenerate in (G, f ) if and only if v is represented by a nonzero vector in every vector representation of (G, f ). 4.3. The signed reduction number Now define the reduction number of a signed graph (G, f ) as R (G, f )
= max{R (G v, f v) + 1 : v nondegenerate in (G, f )}.
If (v1 , . . . , vd ) is a list of vertices of G such that v1 is nondegenerate in (G, f ) and vi+1 is nondegenerate in (G v1 · · · vi , f v1 · · · vi ), then we say that (v1 , . . . , vd ) is a reduction order of (G, f ). Proposition 4.6. Let (G, f ) be a signed graph then R (G, f )
R mr+ (G, f )
and
|G| − Z+ (G) = OS(G) = R (G) R (G, f ). Proof. The first claim follows from Proposition 4.3. We prove the second claim by induction on |G|. When |G| = 1 the claim is clearly true. Now assume |G| > 1. Either R (G) = 0 and we are done or there exists v ∈ V such that R (G v)+ 1 = R (G) and v is nondegenerate in G. Then v is nondegenerate in (G, f ) and so R (G)
= R (G v) + 1 R (G v, f v) + 1 R(G, f ).
Next, define the signed reduction number of a graph G to be R s (G)
= min R (G, f ). f
We then have the first main result of the paper. Theorem 4.7. For a graph G, |G| − Z+ (G)
R (G). = OS(G) = R (G) R s (G) mr+
Proof. The first inequality follows from Proposition 4.6 while the second follows from Equation 2.
5. A natural equivalence on the set of sign patterns of a fixed graph Assume (G, f ) is a signed graph and v ∈ G, then define a new sign pattern Tv f on G by ⎧ ⎨ −f (e) : e incident to v, Tv f (e) = ⎩ f (e) : otherwise.
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Next define an equivalence relation on the sign patterns of G as follows: f
f
= Tv1 Tv2 . . . Tvk g for some v1 , v2 , . . . , vk ∈ G.
∼ g if and only if
The next observation follows directly from the definition of orthogonal removal. Observation 5.1. Suppose (G, f ) is a signed graph with vertex v. Then (Tv f ) v is a vertex of G then Tw (f v) = (Tw f ) v.
= f v and if w = v
R and R depend only on the equivalence As the next proposition shows, the rank parameters mr+ class of f .
Proposition 5.2. Let G be a graph with sign patterns f and g. If f R (G, f ) = R (G, g ).
R (G, f ) = mrR (G, g ) and ∼ g then mr+ +
Proof. Notice that if f ∼ g then PR (G, f ) = D PR (G, g )D−1 where D is a diagonal matrix whose diagR (G, f ) = mrR (G, g ). onal entries consists of 1 and −1. As rank A = rank DAD−1 this implies that mr+ + If f ∼ g and v is a vertex in G, then Observation 5.1 implies that f v ∼ g v. This shows that (v1 , . . . , vd ) is a reduction order of (G, f ) if and only if it is a reduction order of (G, g ) and in particular R (G, f ) = R (G, g ). Proposition 5.3. Let G be a graph with sign pattern f . If T is a simple subtree of G then there exists a sign pattern g of G such that f ∼ g and g |E(T ) ≡ −1. Proof. Induct on |T |. If |T | = 1 then there is nothing to show. Otherwise T has a pendant vertex v. Let e be the unique edge in T incident to v. By induction if T = T − v then there exists a sign pattern g of G such that f ∼ g and g |E(T ) ≡ −1. Now let g = Tv (g ) or g = g depending on the sign of g (e). Proposition 5.3 allows us to pick “nice” representatives of each equivalence class. Example 5.4. Let G be a graph and T be a simple spanning tree of G. Then the set
f : E (G) → {+1, −1} : f |E(T ) ≡ −1 forms a “set of class representatives” of the equivalence ∼ (i.e. each equivalence class is represented exactly once). R (G , f ) 6. Computations for mr+ R (G, f ) for a general graph and a general sign pattern seems very difficult, but in Computing mr+
R (G, f ) for some special cases using the lower bound R (G, f ). this section we compute mr+ R (G, f ) Observation 6.1. For a signed graph (G, f ), mr+
Proof. If A
|G| − 1.
|G| − 1.
∈ PR (G, f ) has smallest eigenvalue λ0 then A − λ0 I|G| ∈ PR (G, f ) and rank(A − λ0 I|G| )
R (T , f ) Corollary 6.2. If T is a simple tree, then mr+
= R (T , f ) = |T | − 1 for any sign pattern f .
R (T , f ). Otherwise T has a pendant vertex v and (T v, f v) Proof. If |T | = 1 then R (T , f ) = 0 = mr+ is a signed simple tree. Then by induction R mr+ (T , f )
R (T , f ) R (T v, f v) + 1 = |T | − 1.
For a graph G = (V , E ) let nG : E → {1, −1} be the negative sign pattern, that is nG (e) ˇ [17] proved the following. all e ∈ E. In the language of vector representations, Šinajová
= −1 for
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Theorem 6.3 [17]. For any connected graph G and sign pattern f
R (G, f ) = R (G, f ) = |G| − 1. ∼ nG , mr+
Proof. The statement is clearly true if |G| = 1. If v ∈ V , then G v is a connected graph on |G| − 1 vertices and nG v is the negative sign pattern on G v. Thus by induction: R mr+ (G, nG )
R (G, nG ) R (G v, nG v) + 1 = |G| − 1.
Proposition 6.4. Let G = (V , E ) be a graph and f , g only one edge, then R R mr+ (G, f ) − mr+ (G, g ) 1.
= (v1 , . . . , vn ) Proof. Let V = g ({v1 , v2 }), then W
: E → {1, −1} be sign patterns of G that differ at
⊂ Rd be a vector representation of (G, f ) of minimum rank. If f ({v1 , v2 })
= (v1 ⊕ (α), v2 ⊕ (1), v3 ⊕ (0), . . . , vn ⊕ (0)) ⊂ Rd+1
R (G, g ) α is chosen correctly. This implies that mr+ R mr+ (G, f ) + 1. Repeating the argument starting with a vector representation of (G, g ) gives the other
is a vector representation of (G, g ) when inequality.
R (G) Theorem 6.5. If mr+
R (G, f ) = d. d |G| − 1 then there exists a sign pattern f such that mr+
R (G) = mrR (G, g ). Proof. By definition there exists a sign pattern g : E → {1, −1} such that mr+ + Construct a sequence of sign patterns g = g0 , g1 , . . . , gm = nG such that gi and gi+1 differ at exactly one edge. Then by Proposition 6.4 and Proposition 6.3 R R R R {mr+ (G, g0 ), . . . , mr+ (G, gm )} = {mr+ (G), mr+ (G) + 1, . . . , |G| − 1}.
1
For a signed simple cycle (Cn , f ) with vertices V (Cn ) = {1, . . . , n} and edges E (Cn ) = {{i, i + 1} : i n} we define the parameter: sign(Cn , f )
= (−1)n−1
n
i=1
f ({i, i + 1}).
Proposition 6.6. With the above notation, for n 3 ⎧ ⎨ n − 2 : if sign(C , f ) n R mr+ (Cn , f ) = R (Cn , f ) = ⎩ n − 1 : if sign(C , f ) n
= 1, = −1.
Proof. It is clear that if f ∼ g then sign(Cn , f ) = sign(Cn , g ). Further there are exactly two equivalence classes of sign patterns and sign(Cn , nCn ) = −1. The proposition then follows from Proposition 5.2, K (C ) Theorem 6.3, Theorem 6.5, and the fact that mr+ n
= n − 2.
7. Proof of Theorem 1.4 7.1. Graphs for which |G| − Z+ (G)
R (G) = R (G) = OS(G) < mr+
K (G) > R (G) come from the Möbius ladder on 8 vertices A major source of examples for which mr+ (see Fig. 1) which we denote ML8 . Mitchell et al. [14] showed that
5
K = mr+ (ML8 ) > R (ML8 ) = 4.
(5)
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K (G) by considering vertex sums Mitchell et al. constructed additional graphs for which R (G) < mr+ and joins of graphs. A vertex v in a connected graph G = (V , E ) is a cut-vertex if G[V − v] is disconnected. A maximal connected induced subgraph without a cut-vertex is called a block. Booth et al. [3] and van der Holst [18] have independently shown that the positive semidefinite minimum rank is completely determined by the blocks of G independently of how they are joined. More specifically: if G1 , ..., Gk are the blocks K (G) = k mrK (G ). The reduction number, like the positive semidefinite minimum of G then mr+ + i i=1 rank, behaves nicely over the blocks of G.
Theorem 7.1 [14, Theorem 4.1]. If G1 , ..., Gk are the blocks of a simple graph G then R (G)
=
k
i=1
R (Gi ).
A graph G is a vertex sum of two graphs G1 and G2 if G is the superposition of G1 and G2 identified at a vertex v. By letting G be a vertex sum of N copies of ML8 we see that 4N
K = R (G) < mr+ (G) = 5N
K (G) − R (G) can be arbitrarily large. showing that mr+ Given two graphs G1 and G2 the join of G1 and G2 , denoted G1 V (G1 ) ∪ V (G2 ) and edge set E (G1 ) ∪ E (G2 ) ∪ E where
E
∨ G2 , is the graph with vertex set
= {uv : u ∈ V (G1 ) and v ∈ V (G2 )}.
C (G ∨ G ) = max{mrC (G ∨ K ), mrC (G ∨ K )}. Barioli et Hackney et al. [10] showed that mr+ 1 2 1 1 + 1 + 2 al. [2] proved this in the real case. The reduction number also behaves like the positive semidefinite minimum rank when considering joins of graphs.
Theorem 7.2 [14, Theorem 4.3]. If G1 and G2 are connected simple graphs on two or more vertices, then R (G1
∨ G2 ) = max{R (G1 ), R (G2 )}.
Using Theorem 7.2, Mitchell et al. constructed for each N such that K mr+ (Gt ) − R (Gt )
> 0 an infinite number of graphs {Gt }t 2
= N.
For example one can set Gt = H ∨ Kt where H is a graph with exactly N blocks each identical to ML8 and Kt is the complete graph on t vertices. 7.2. Vertex sums and graph joins for the parameter R s (G) R (ML ) = 5 and state variants of Theorem 7.1 and 7.2 for We will next show that R s (ML8 ) = mr+ 8 the signed reduction number. This will prove Theorem 1.4. We begin with a lemma.
Lemma 7.3. Suppose (G, f ) is a signed graph. If H is an induced subgraph of G and h R (H , h) R (G, f ). In particular, R s (H ) R s (G).
= f |E(H ) then
Proof. Induct on R (H , h). If R (H , h) = 0 then the lemma follows. Now suppose R (H , h) > 0, then there exists a nondegenerate vertex v in (H , h) such that R (H , h) = R (H v, h v) + 1. As H is an induced subgraph of G, v is also nondegenerate in (G, f ). Further H v is an induced subgraph of G v and h v = (f v)|E(H v) so induction implies that R (H , h)
= R (H v, h v) + 1 R (G v, f v) + 1 R (G, f ).
Proposition 7.4. R s (ML8 )
= 5.
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Fig. 1. The graph ML8 and the graph ML8 with some distinguished edges.
Proof. Fix a sign pattern f of ML8 . We wish to show that R (ML8 , f ) replace f with an equivalent sign pattern (see Fig. 1) such that f (12)
5. Using Proposition 5.3 we can
= f (23) = f (26) = f (34) = f (37) = f (45) = f (48) = −1. = G[{1, 2, 3, 4, 5, 7}] and h = f |E(H ) then h ≡ −1.
Case 1: Suppose f (15) = −1. In this case, if H So Theorem 6.3 and Lemma 7.3 implies that R (G, f )
R (H , h) = |H | − 1 = 5.
Case 2: Suppose f (18) = −1 or f (56) = −1. Via the symmetry of ML8 we may suppose f (18) = −1. In this case, if H = G[{1, 2, 3, 4, 6, 8}] and h = f |E(H ) then h ≡ −1. So Theorem 6.3 and Lemma 7.3 implies that
R (H , h) = |H | − 1 = 5. Case 3: Suppose f (78) = −1 or f (67) = −1. Via the symmetry of ML8 we may suppose f (78) = −1. In this case, if H = G[{2, 3, 4, 5, 7, 8}] and h = f |E(H ) then h ≡ −1. So Theorem 6.3 and R (G, f )
Lemma 7.3 implies that
R (H , h) = |H | − 1 = 5. Case 4: Suppose f (15) = f (18) = f (56) = f (67) = f (78) = +1. In this case, if H = G[{1, 3, 4, 5, 6, 7}] and h = f |E(H ) then T1 T6 (h) ≡ −1. So Theorem 6.3 and Lemma 7.3 implies R (G, f )
that
R (G, f )
R (H , h) = |H | − 1 = 5.
Like the standard reduction number and OS-number, the signed reduction number is additive over the blocks of G. Theorem 7.5. If G1 , ..., Gk are the blocks of a simple graph G then R s (G)
=
k
i=1
R s (Gi ).
Theorem 7.5 follows from the next Lemma. Lemma 7.6. If v ∈ V and (G, f ) is simple and connected, then there exists a maximal signed reduction / S. order S of (G, f ) such that v ∈ The proof of Lemma 7.6 is somewhat involved so we delay it until the next section. Proof of Theorem 7.5. It is enough to show that R (G, f ) = R (G1 , f1 ) + R (G2 , f2 ) if G is a vertex sum of G1 and G2 at the vertex v and fi = f |E(Gi ) . Let S be a maximal reduction order of (G, f ) which does not contain the vertex v. Then Si = S ∩ V (Gi ) is a reduction order for (Gi , fi ) and so R (G, f )
= |S| = |S1 | + |S2 | R (G1 , f1 ) + R (G2 , f2 ).
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To see the other inequality, let Si be a maximal reduction order of (Gi , fi ) not containing the vertex v. Then S = S1 S2 is a reduction order of (G, f ) showing that R (G, f )
|S| = |S1 | + |S2 | = R (G1 , f1 ) + R (G2 , f2 ).
Corollary 7.7. The set Es is closed under vertex sums. Lemma 7.8. If G is simple and G
∈ Es then G ∨ v ∈ Es .
Proof. If G is disconnected and G1 , . . . , Gk are the connected components of G then v is a cut vertex R (G ∨ v) = mrR (G ∨ v) and R (G ∨ v) = R (G ∨ v). Hence we may of G ∨ v and so mr+ s s i + i assume that G is connected. If |G| = 1, then G ∨ v = K2 and the lemma is clearly true. Hence we may assume that G has two or more vertices. Then as G is connected and has two or more vertices R (G) = mrR (G ∨ v) [10, Corollary 2.7] and mr+ + R s (G)
R R = mr+ (G) = mr+ (G ∨ v) R s (G ∨ v) R s (G).
Notice that the last inequality follows from Lemma 7.3. Corollary 7.9. The set Es is closed under graph joins. Proof. Let G1 , G2 R (G1 mr+
Further as Gi
∈ Es then by Lemma 7.8,
R R ∨ G2 ) = max{mr+ (G1 ∨ v), mr+ (G2 ∨ v)} = max{R s (G1 ∨ v), R s (G2 ∨ v)}.
∨ v is an induced subgraph of G1 ∨ G2 , Lemma 7.3 implies that
max{R s (G1
R ∨ v), R s (G2 ∨ v)} R s (G1 ∨ G2 ) mr+ (G1 ∨ G2 ).
8. Proof of Lemma 7.6 The proof of Lemma 7.6 follows the proof of the corresponding result for OS-sets in [14, Proposition 2.16, Corollary 2.17], but the technical details in the signed case make the argument much more complicated. As in [14], we begin the proof of the lemma with two observations. Observation 8.1. If (v1 , . . . , vd ) is a reduction order of (G, f ) and vd is adjacent to wd in G v1 · · · vd−1 by edges of all the same sign, then (v1 , . . . , vd−1 , wd ) is a reduction order of (G, f ). Observation 8.2. Let (v1 , . . . , vd ) be a reduction order of (G, f ). If σ is a permutation of {1, . . . , d} such that σ (i) < σ (j) whenever i < j and vi and vj are in the same connected component of G[{v1 , . . . , vd }] then (vσ (1) , . . . , vσ (d) ) is a reduction order of (G, f ). Lemma 8.3. If (v1 , . . . , vd ) is list of vertices in (G, f ) and v is adjacent to w in (G, f ) v1 · · · vd by all edges of the same sign then v and w are either not adjacent or adjacent by all edges of the same sign in (G, f ) v1 · · · vβ−1 vβ+1 · · · vd . Proof. Let (G , f ) = (G, f ) v1 · · · vβ−1 and notice that (G , f ) − vβ vβ+1 · · · vd = (G , f ) vβ+1 · · · vd − vβ . In particular, it is enough to show that v and w are either not adjacent or adjacent by all edges of the same sign in (G , f ) − vβ vβ+1 · · · vd . This is easy to see: as (G , f ) − vβ is a sub sign pattern of (G , f ) vβ , the definition of orthogonal removal implies that (G , f ) − vβ vβ+1 · · · vd is a sub sign pattern of (G , f ) vβ vβ+1 · · · vd .
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Before stating the next lemma, we will need some notation. Notice that the map
(G, f ) → PR (G, f ) is not injective, many different signed graphs can determine the same set of positive semidefinite matrices. Because of this we will we introduce the concept of matrix equivalence between signed graphs. Consider the following two operations on a signed graph: (O1) Suppose (G, f ) is a signed graph and e is an edge in G incident to vertices x and y. Construct a new signed graph (G , f ) by adding an edge e to G incident to x and y such that f (e ) = f (e). e both incident to vertices x and y. If (O2) Suppose (G, f ) is a signed graph with edges e and f (e) = f (e ), construct a new signed graph G, f by removing the edge e from G. Definition 8.4. We say two signed graphs (G1 , f1 ) and (G2 , f2 ) are matrix equivalent and write (G1 , f1 ) ≈ (G2 , f2 ) if (G1 , f1 ) can be constructed from (G2 , f2 ) by repeated applications of operations (O1) and (O2) above. The motivation for the words “matrix equivalence” comes from the first part of the next observation. Observation 8.5. Suppose (G1 , f1 ) and (G2 , f2 ) are signed graphs. Then (1) (G1 , f1 ) ≈ (G2 , f2 ) if and only if PR (G1 , f1 ) = PR (G2 , f2 ), (2) If (G1 , f1 ) ≈ (G2 , f2 ) and v is a vertex in V (G1 ) = V (G2 ) then
(G1 , f1 ) v ≈ (G2 , f2 ) v. In general we cannot expect that G v u
= G u v, but we can prove the following.
Lemma 8.6. Suppose (G, f ) is a signed graph with vertices v and u. If u and v are either not adjacent or adjacent by edges of all the same sign in (G, f ), then
(G, f ) u v ≈ (G, f ) v u. Example 8.7. The assumption on u and v is necessary in Lemma 8.6. Take for instance the graph H1 in Fig. 2. Let e be the edge between u and v. Add another edge e between u and v and suppose f is a sign pattern such that f (e) = −1, f (e ) = 1, f (vx) = −1, and f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by one edge with sign −1. While in (G, f ) v u the vertices x and y are adjacent by five edges, two of these edges have sign +1 while three have sign −1. Hence, in this case, (G, f ) u v and (G, f ) v u are not matrix equivalent. Proof of Lemma 8.6. We begin by making a series of reductions: (1) if x, y are vertices of G and H = G[u, v, x, y] then (H , f ) v u = (G, f ) v u [x, y] and
(H , f ) u v = (G, f ) u v [x, y] hence we may assume |G| = 4 and V (G) = {x, y, u, v}, (2) we may assume u and v are adjacent in G, for otherwise it is clear that
(G, f ) v u = (G, f ) u v, (3) using reduction 1, we may assume {x, y} ⊂ N (v) ∪ N (u) for otherwise it is clear that (G, f ) v u = (G, f )[x, y] = (G, f ) u v,
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Fig. 2. The graphs H1 , H2 , H3 , and H4 .
(4) we may assume that x and y are not adjacent in G, for suppose e is an edge in G incident to x and y then ((G, f ) − e) v u = (G, f ) v u − e and ((G, f ) − e) u v = (G, f ) u v − e. So ((G, f ) − e) v u ≈ ((G, f ) − e) u v if and only if (G, f ) v u ≈ (G, f ) u v. With the reductions above and by possibly switching u and v or x and y, the underlying simple graph of G appears in Fig. 2. Now direct inspection shows that if two vertices in G are adjacent by edges of differing signs then x and y are adjacent by edges of differing signs in (G, f ) v u and (G, f ) u v. 1 Hence, in this case,
(G, f ) v u ≈ (G, f ) u v. So by replacing (G, f ) with a matrix equivalent signed graph we may suppose G is simple. Then using the graphs in Fig. 2 we have the following four cases. Case 1: G = H1 . By Observation 5.1, we may suppose that f (vu) = f (vx) = f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by one edge with sign −1. In (G, f ) v u, the vertices x and y are adjacent by two edges both with sign −1. Hence, in this case,
(G, f ) v u ≈ (G, f ) u v. Case 2: G = H2 . By Observation 5.1, we may suppose that f (yu) = f (uv) = f (vx) = −1. Then in both (G, f ) u v and (G, f ) v u, the vertices x and y are adjacent by one edge with sign −1. Hence, in this case,
(G, f ) v u = (G, f ) u v. Case 3: G = H3 . By Observation 5.1, we may suppose that f (vu) = f (vx) = f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by two edge with signs −1, f (ux). In (G, f ) v u, the vertices x and y are adjacent by three edges with signs −1, −1, f (ux). Hence, in this case,
(G, f ) v u ≈ (G, f ) u v. Case 4: G = H4 . By Observation 5.1, we may suppose that f (vu) = f (vx) = f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by five edge with signs −1, f (ux), f (uy), −f (xu)f (uy), −f (xu)f (uy). In (G, f ) v u, the vertices x and y are adjacent by five edges with signs −1, −1, f (ux), f (uy), −f (xu)f (uy). Hence, in this case, (G, f ) v u ≈ (G, f ) u v. This completes the proof. Lemma 8.8. Let (v1 , . . . , vd ) be a reduction order of a signed graph (G, f ) such that vd is adjacent to wd in G v1 · · · vd−1 by all edges of the same sign. If C is a cycle in G[{v1 , . . . , vd } ∪ {wd }] containing the vertices vβ , vd , wd then
(v1 , . . . , vβ−1 , vβ+1 . . . , vd , wd ) and (v1 , . . . , vβ−1 , vβ+1 , . . . , vd , vβ ) are both reduction orders of (G, f ). 1 Notice that this assertion uses the fact that u and v are adjacent by all edges of the same sign, otherwise the last assertion would fail when G = H1 , see Example 8.7.
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Remark 8.9. When we say C is a cycle in H, we mean that C consists a list of distinct vertices c1 , . . . , cN in H and a list of edges e1 , . . . , eN such that ei is incident to ci and ci+1 (where cN +1 = c1 ). We do not assume C is an induced graph of H. Notice that a cycle on 2-vertices will be two vertices adjacent by / {c1 , . . . , cN } two edges. Also observe that if w ∈ H then there is a natural cycle C w in H w. If w ∈ we let C w = C. If w = ck for some k, then we let C w be the simple cycle with vertices c1 , . . . , ck−1 , ck+1 , . . . , cN and edges e1 , . . . , ek−2 , e , ek+1 , . . . , eN where e is the edge in H w coming from the edges ei−1 and ei in H (see Section 4.2). Proof of Lemma 8.8. Let
(Gk , fk ) = (G, f ) v1 · · · vβ+k and
(Hk , hk ) = (G, f ) v1 · · · vβ−1 vβ+1 · · · vβ+k . We will first prove the following claim. Claim: For 0 k d − β , (Hk , hk ) vβ ≈ (Gk , fk ). Induct on k. For k = 0, (H0 , h0 ) vβ claim holds for some k < d − β . Then
= (G0 , f0 ) and so the base case is true. Now suppose the
(Hk+1 , hk+1 ) vβ = (Hk , hk ) vβ+k+1 vβ and by Observation 8.5 and the induction hypothesis
(Gk+1 , fk+1 ) = (Gk , fk ) vβ+k+1 ≈ (Hk , hk ) vβ vβ+k+1 . In particular, if vβ and vβ+k+1 are not adjacent or adjacent by all edges of the same sign in Hk then Lemma 8.6 implies that
(Hk+1 , hk+1 ) vβ ≈ (Gk+1 , fk+1 ). Assume for a contradiction that vβ and vβ+k+1 are adjacent by edges of differing sign in Hk . By the hypothesis of the lemma and Remark 8.9 there exists a simple cycle C = vd wd c1 . . . c vβ c+1 . . . cN vd containing vd , wd , and vβ in Hk such that {c1 , . . . , cN } ⊂ {vk+β+1 , . . . , vd }.
/ {c1 , . . . , cN }. Then by the definition of orthogonal removal, c is adjacent Case 1: Suppose vβ+k+1 ∈ to vβ+k+1 by edges of differing signs in Hk vβ . The same is true for c+1 . Hence c and c+1 are adjacent to each other by edges of differing signs in Hk vβ vβ+k+1 . Now let σ be a permutation of {1, . . . , N } such that (cσ (1) , cσ (2) , . . . , cσ (N ) ) is sublist of (vk+β+2 , . . . , vd ). Then as (Hk , hk ) vβ ≈ (Gk , fk ), by Lemma 8.3 and Observation 8.5 the vertices vd and wd must either be nonadjacent or adjacent by all edges of the same sign in (Hk , hk ) vβ vβ+k+1 cσ1 · · · cσN . However, as c and c+1 are adjacent by edges of differing sign in (Hk , hk ) vβ vβ+k+1 and C vβ vβ+k+1 is a cycle containing c , c+1 , vd and wd this is impossible. So we have a contradiction and the proof of Claim 1 is complete in this case. Case 2: Suppose vβ+k+1 ∈ {c1 , . . . , cN }. Then by replacing C with a shorter cycle we may assume that vβ+k+1 is one of c or c+1 . Then by the definition of orthogonal removal, c and c+1 are adjacent to each other by edges of differing signs in Hk vβ . Now the same argument as above implies a contradiction. This completes the proof of the Claim. Now for 0
k d − β − 1 we have
(Gk , fk ) ≈ (Hk , hk ) vβ ,
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so vβ+k+1 is nondegenerate in (Hk , hk ) vβ and hence also nondegenerate in (Hk , hk ). Thus (v1 , . . . , vβ−1 , vβ+1 , . . . , vd−1 ) is a reduction order of (G, f ). So if
(H , h) = (G, f ) v1 · · · vβ−1 vβ+1 · · · vd−1 we must show that (vd , wd ) and (vd , vβ ) are both reduction orders in (H , h). Let (B, b) = (H , h) [vd , wd , vβ ], by the hypothesis of the lemma the underlying simple graph of B is the 3-cycle. Further, as vd and wd are adjacent by edges of the all the same sign in
(G, f ) v1 · · · vd = (Gd−β−1 , fd−β−1 ) ≈ (H , h) vβ we see that R (B, b) = 2. This implies, by Proposition 6.6, that b is equivalent to the all negative sign pattern. Then both (vd , wd ) and (vd , vβ ) are reduction orders of (B, b) and hence (H , h). Proof of Lemma 7.6. If there exists a maximal reduction order S of (G, f ) such that v ∈ / S we are done. So assume for a contradiction that every maximal reduction order of (G, f ) contains the vertex v. Then for a reduction order S = (u1 , . . . , ud ) of (G, f ) define the number NS to be the number of vertices uj picked after v such that uj and v are in the same connected component of G[S ]. If there exists a maximal reduction order S with NS = 0, then by Observation 8.2 we may rearrange our reduction order so that v is picked last. Then by Observation 8.1 there exists a reduction order S not containing v. Hence by our assumption, NS > 0 for every maximal reduction order. Let S = (u1 , . . . , ud ) be a reduction order such that NS is minimal. By Observation 8.2 we may assume that ud and v are in the same connected component of G[u1 , . . . , ud ]. Further, there exists a vertex wd adjacent to ud in G u1 · · · ud−1 by all edges of the same sign. We will construct a reduction order S such that NS < NS , thus contradicting the minimality of NS . Case 1: wd and v are in different connected component of G[{wd } ∪ {u1 , . . . , ud−1 }]. Then by Observation 8.1, S = (u1 , . . . , ud−1 , wd ) is a reduction order and NS < NS . Case 2: wd and v are in the same connected component of G[{wd } ∪ {u1 , . . . , ud−1 }]. Let H = G[{wd } ∪ {u1 , . . . , ud }]. By our assumptions, there exists a path P1 connecting v to wd in H that does not contain the vertex ud . As v and ud are in the same connected component of G[{u1 , . . . , ud }], there also exists a path P2 connecting v to ud in H that does not contain the vertex wd . Suppose v = uk and then using these paths we have the following two cases. Case 2(a): There is a simple cycle C in H containing wd , v, and ud . Then by Lemma 8.8, S = (u1 , . . . , uk−1 , uk+1 , . . . , ud , v) is a reduction order of (G, f ) and NS = 0. Case 2(b): There exists a simple cycle C in H containing wd , ud and a vertex uj such that wd and v are in different connected components of G[{wd } ∪ {u1 , . . . , ud } \ {uj }]. Then by Lemma 8.8, S = (u1 , . . . , uˆj , . . . , ud , wd ) is a reduction order of (G, f ) and NS < NS . Acknowledgements The author would like to thank Sivaram K. Narayan for his helpful comments on an earlier draft of this paper and the referee for pointing out an error in the original “proof” of Lemma 7.6. References [1] Francesco Barioli, Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Bryan Shader, P. van den Driessche, Hein van der Holst, Zero forcing parameters and minimum rank problems, Linear Algebra Appl. 433 (2) (2010) 401–411. [2] Francesco Barioli, Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Hein van der Holst, On the graph complement conjecture for minimum rank, Linear Algebra Appl. 436 (12) (2012) 4373–4391., Special Issue on Matrices Described by Patterns. [3] Matthew Booth, Philip Hackney, Benjamin Harris, Charles R. Johnson, Margaret Lay, Terry Lenker, Lon H. Mitchell, Sivaram K. Narayan, Amanda Pascoe, Brian D. Sutton, On the minimum semidefinite rank of a simple graph, Linear and Multilinear Algebra 59 (5) (2011) 483–506. [4] Matthew Booth, Philip Hackney, Benjamin Harris, Charles R. Johnson, Margaret Lay, Lon H. Mitchell, Sivaram K. Narayan, Amanda Pascoe, Kelly Steinmetz, Brian D. Sutton, Wendy Wang, On the minimum rank among positive semidefinite matrices with a given graph, SIAM J. Matrix Anal. Appl. 30 (2) (2008) 731–740.
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[5] Luz Maria DeAlba, Timothy L. Hardy, Irvin Roy Hentzel, Leslie Hogben, Amy Wangsness, Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2–3) (2006) 394–415. [6] Jason Ekstrand, Craig Erickson, Diana Hay, Leslie Hogben, Jolie Roat, Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees, Electron. J. Linear Algebra 23 (2012) 79–87. [7] Shaun M. Fallat, Leslie Hogben, The minimum rank of symmetric matrices described by a graph: a survey, Linear Algebra Appl. 426 (2–3) (2007) 558–582. [8] Shaun M. Fallat, Leslie Hogben, Variants on the minimum rank problem: a survey II, preprint, 2011. [9] AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (7) (2008) 1628–1648. [10] Philip Hackney, Benjamin Harris, Margaret Lay, Lon H. Mitchell, Sivaram K. Narayan, Amanda Pascoe, Linearly independent vertices and minimum semidefinite rank, Linear Algebra Appl. 431 (8) (2009) 1105–1115. [11] Frank J. Hall, Zhongshan Li, Sign pattern matrices, in: Leslie Hogben (Ed.), Handbook of Linear Algebra, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2007 [12] Leslie Hogben, A note on minimum rank and maximum nullity of sign patterns, Electron. J. Linear Algebra 22 (2011) 203–213. [13] Satyanarayana V. Lokam, Complexity lower bounds using linear algebra, Foundations and Trends® in Theoretical Computer Science 4(1–2):front matter, 1–155 (2009), 2008. [14] Lon H. Mitchell, Sivaram K. Narayan, Andrew M. Zimmer, Lower bounds in minimum rank problems, Linear Algebra Appl. 432 (1) (2010) 430–440. [15] Lon H. Mitchell, Sivaram K. Narayan, Andrew M. Zimmer, Lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs, Linear Algebra Appl. 436 (3) (2012) 525–536. [16] T.D. Parsons, Tomaž Pisanski, Vector representations of graphs, Discrete Math. 78 (1–2) (1989) 143–154. [17] Edita Šinajová, ˇ A note on vector representation of graphs, Discrete Math. 89 (3) (1991) 315–317. [18] Hein van der Holst, On the maximum positive semi-definite nullity and the cycle matroid of graphs, Electron. J. Linear Algebra 18 (2009) 192–201.
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Linear Algebra and its Applications journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / l a a
A new lower bound for the positive semidefinite minimum rank of a graph Andrew M. Zimmer Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States
ARTICLE INFO
ABSTRACT
Article history: Received 26 June 2011 Accepted 9 September 2012 Available online 6 October 2012
The real positive semidefinite minimum rank of a graph is the minimum rank among all real positive semidefinite matrices that are naturally associated via their zero-nonzero pattern to the given graph. In this paper, we use orthogonal vertex removal and sign patterns to improve the lower bound for the real positive semidefinite minimum rank determined by the OS-number and the positive semidefinite zero forcing number. © 2012 Elsevier Inc. All rights reserved.
Submitted by S. Fallat AMS classification: 05C50 15A03 15A18 15B35 15B57 Keywords: Positive semidefinite minimum rank Minimum rank Sign patterns Zero forcing number Positive semidefinite zero forcing number Ordered set number
1. Introduction A graph G = (V , E ) consists of a set of vertices V and a set of edges E, where the elements of E are unordered pairs of vertices. The order of G, denoted |G|, is the cardinality of V . A graph is simple if it has no multiple edges or loops. The entries of an n-by-n Hermitian matrix A = (aij ) over the complex numbers C naturally correspond to a simple undirected graph G(A) with vertex set {1, . . . , n} and edge set {{i, j} : aij = 0, 1 i < j n}. Observe that the diagonal entries of A have no impact on the structure of G(A). E-mail address: [email protected] 0024-3795/$ - see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2012.09.001
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Within this correspondence a number of extremal problems arise, one of recent interest is the minimum rank problem, which seeks to determine the smallest possible rank of any real symmetric matrix associated to a given graph (see the 2007 survey by Fallat and Hogben [7]). Define S (G)
= {A ∈ Mn (R) : AT = A, G(A) = G}.
The minimum rank of G is defined to be mr(G)
= {rank(A) : A ∈ S (G)}.
In this paper, we consider the related problem of determining the minimum rank among positive semidefinite (psd) matrices associated to a given graph (see the 2011 survey by Fallat and Hogben [8]). For K ∈ {R, C} define PK (G)
= {A ∈ Mn (K) : A psd, G(A) = G}.
The real or complex positive semidefinite minimum rank of G is defined to be K mr+ (G)
= {rank(A) : A ∈ PK (G)}.
The real and complex parameters are different, for instance Barioli et al. [1] constructed a graph G for C (G) < mrR (G). which mr+ + A major problem when considering these rank parameters is developing combinatorial conditions K (G) or mr(G) from above or below. Along these lines, the on the graph G = (V , E ) that bound mr+ notion of a zero forcing set was introduced in [9] to bound mr(G) from below. A set Z ⊂ V is a zero forcing set if it satisfies a combinatorial condition involving vertex coloring. One then defines the zero forcing number, denoted Z (G), to be the cardinality of the smallest zero forcing set. By studying the support of vectors in the null space of any matrix representation of G one can show that |G| − Z(G) is a lower bound for mr(G). Theorem 1.1 [9, Proposition 2.4]. |G| − Z(G)
mr(G).
K (G). Hackney et al. [10] introduced the notion of an OS-set to construct a lower bound for mr+ A set S ⊂ V is a OS-set if it satisfies a combinatorial condition involving connectivity in various induced subgraphs. The OS-number, OS(G), is the cardinality of the largest OS-set. By considering K (G). vector representations of a graph G one can show that OS(G) is a lower bound for mr+
Theorem 1.2 [10, Proposition 3.3]. OS(G)
K (G). mr+
Although the two lower bounds were developed independently, it turns out that they are almost “dual” to one another. Mitchell et al. [14, Theorem 2.10] showed that if Z is a zero forcing set then V \ Z is an OS-set. The reverse is not always true: if S is an OS-set we have no guarantee that V \ S is a zero forcing set. However Mitchell et al. defined an OS-number like parameter which equals |G| − Z(G). Shortly after Barioli et al. [1] defined a zero forcing number like parameter called the positive semidefinite zero forcing number, Z+ (G), such that
|G| − Z+ (G) = OS(G). C (G) Hackney et al. [10] showed that mr+
= |G| − Z+ (G) for all chordal graphs and all graphs on six C (G) for all graphs. or fewer vertices. Based on this evidence they conjectured that |G| − Z+ (G) = mr+ But Mitchell et al. [14] constructed examples for which K (G) − OS(G) mr+
K = mr+ (G) − (|G| − Z+ (G))
can be arbitrarily large. In this paper we specialize to the real case and introduce a new lower bound R (G) called the signed reduction number, denoted by R (G). We obtain the following relationship for mr+ s with the previously defined parameters.
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Theorem 1.3. |G| − Z+ (G)
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R (G). = OS(G) R s (G) mr+
We further show that |G| − Z+ (G) = OS(G) In particular we prove the following theorem.
R (G) for the graphs considered in [14]. < R s (G) = mr+
R (G)} and E = {G simple graph : R (G) = = {G simple graph : |G| − Z+ (G) = mr+ s s
Theorem 1.4. Let E
R (G)}. Then mr+
(1) E ⊂ Es , (2) ML8 ∈ Es \ E, (3) both E and Es are closed under vertex sums and graph joins. Here ML8 is the Möbius ladder on 8 vertices and the operations of vertex sums and graph joins are / E and E is closed under vertex sums and graph joins follows given in Section 6. The fact that ML8 ∈ from the work in [14]. The motivation for the word “reduction” comes from a new interpretation of the OS-number developed by Mitchell et al. [15] and described in Section 3 of this paper. The motivation for the word “signed” follows from the use of sign patterns to construct R s (G). Given a simple graph G = (V , E ), a sign pattern of G is a function f : E → {1, −1}. We call the pair (G, f ) a signed graph. If (G, f ) is a signed graph then PR (G, f ) ⊂ PR (G) denotes the set of all real psd matrices A = (aij ) such that:
• aij < 0 if ij ∈ E and f (ij) = −1, • aij > 0 if ij ∈ E and f (ij) = 1, • aij = 0 if ij ∈ / E. The positive semidefinite minimum rank of a signed graph (G, f ) is defined to be R mr+ (G, f )
= min{rank A : A ∈ PR (G, f )}.
In Section 4, we construct a lower bound for signed graphs, denoted by R (G, f ), such that R |G| − Z+ (G) R (G, f ) mr+ (G, f ).
(1)
By exploiting the fact that R mr+ (G)
R = min mr+ (G, f ) f
(2)
R (G), namely one obtains a new lower bound for mr+
R s (G)
= min R (G, f ). f
By the inequality in 1, we then have R |G| − Z+ (G) R s (G) mr+ (G).
1.1. Some history The basic problem of understanding how the zero-nonzero pattern or sign pattern of a matrix A determines its spectral properties is an old and well studied problem. In this short subsection, we will mention some previous work involving sign patterns and minimum rank. A more in depth introduction can be found in [8, Section 5.3] or [11, Chapter 33]. Given a signed simple graph (G, f ) one can naturally define S (G, f ) ⊂ S (G) and the minimum rank parameter mr(G, f ). With slightly different notation, mr(G, f ) has been computed when G is a tree [5,12]. Several authors have also considered a non-symmetric variant of this parameter: given a
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m-by-n sign pattern Z, what is the minimum rank of a matrix A with sign pattern Z? In this context, a number of authors have explored interesting connections between the minimum rank of full sign patterns and communication complexity, for additional information see the book by Lokam [13]. R (G, f ) will behave very differently than the corresponding minimum We suspect the parameter mr+ rank parameter mr(G, f ). One key difference is that the set S (G) is closed under additive inverses: if A ∈ S (G) then −A ∈ S (G). This “closure” does not hold for the set PR (G). As a result of this difference, the main results of Section 5 of this paper have no minimum rank analogue. For instance if Km is the complete graph on m-vertices and n : E (Km ) → {+1, −1} is the all negative sign pattern then R (K , n) = m − 1 (see Theorem 6.3). mr(Km , n) = 1 but mr+ m In the language of vector representations, Parson and Pisanski [16] and Šinajová ˇ [17] considered R (G, n) when n is the all negative sign pattern: n(e) = −1 for all e ∈ G. the problem of computing mr+
R (G, n) = |G| − 1 when G is connected. In Section 5 we will In this case, Šinajová ˇ [17] proved that mr+ prove this result using the parameter R (G, n). The central idea of this paper, which does not appear to be present in any of the papers cited above, is that sign patterns can be exploited to obtain better bounds on the minimum rank of a set of matrices determined by a fixed zero-nonzero pattern.
1.2. Outline of paper Section 2 is devoted to some preliminaries. In Section 3 we describe the reduction number. Section 4 is the beginning of our new results, there we refine the reduction number for sign patterns. In Section 5 we describe a natural equivalence relation on the set of sign patterns of a fixed graph. In Section 6 we R (G, f ) for some special cases including when G is a cycle or tree. In Section 7 we prove compute mr+ Theorem 1.4. Finally Section 8 is devoted to proving a lemma used in Section 7. 2. Preliminaries 2.1. Graph theory As our only concern is undirected graphs, an edge {v, u} will sometimes be written as vu. The subgraph G[R] of G induced by R ⊂ V (G) is the subgraph of G with vertex set R and edge set consisting of those edges of G where both vertices are elements of R. For a vertex w of a graph G, let N (w) denote the set of all vertices adjacent to w in G, called the neighborhood of w in G. By the closed neighborhood of w, denoted N [w], we mean {w} ∪ N (w). Let Kn denote the complete graph on n vertices, Cn the cycle on n vertices, and Pn the path on n vertices. If H and K are graphs, let H K denote the disjoint union of H and K. If G is a graph let nG
= G · · · G . n copies
2.2. Vector representations
Given a list of n column vectors in Kd , X
= (x1 , . . . , xn ), let X be the matrix [x1 . . . xn ]. Then X ∗ X
with regard to the Euclidean inner product. Its associated is a psd matrix called the Gram matrix of X x1 , . . . , xn , and edges corresponding to simple graph G has n vertices corresponding to the vectors is a vector nonzero inner products among those vectors. Since X ∗ X ∈ PK (G) for the graph G, we say X , we mean the dimension of the span of the vectors in X , which is equal to representation of G. By rank X rank X ∗ X. As every positive semidefinite matrix is a Gram matrix and conversely every Gram matrix K (G) d if and only if there is a vector representation of G in Kd . is positive semidefinite, mr+ This type of vector representation is a particular instance of the more general vector representations studied by Parson and Pisanski [16].
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2.3. Multigraphs A graph G is a multigraph if G is an undirected graph that has no loops but may have multiple edges between vertices. If G is a multigraph then PK (G) denotes the set of all psd matrices with entries in K where:
• aij = 0 if vertex i and vertex j are connected by exactly one edge in G, • aij = 0 if vertex i and vertex j are not adjacent and i = j. No restriction is placed on aij if vertex i and vertex j are connected by more than one edge. The (real or complex) positive semidefinite minimum rank of a multigraph G is defined as: K mr+ (G)
= min{rank A : A ∈ PK (G)}.
Notice that vector representations generalize to multigraphs in an obvious way. 2.4. Orthogonal vertex removal We next present the concept of orthogonal vertex removal. This was introduced by Booth et al. [4], but we will follow the slightly different but equivalent formulation in Mitchell et al. [15]. Consider ⊂ Kd of a graph G = (V , E ). For v ∈ V with vector representative v a vector representation V consider the subspace U = Span{ v} and the orthogonal projection Pv : Kd → U ⊥ onto its orthogonal form a vector representation of a new simple graph G on n − 1 vertices complement. The vectors Pv V element). v=0 (we ignore the Pv To understand the graph G consider the inner product:
Pv u, Pv w = u, w −
u, v v, w v, v
for some u, w ∈ V and their corresponding vector representatives u , w. The expression above shows that G could be one of several possibilities. To model this we use multigraphs. If v ∈ V (G), the orthogonal removal of v from G, denoted G v, is the multigraph modified from the induced subgraph on the vertices V (G) − {v} such that any u, w ∈ N (v) have P additional edges added between them, where P is the product of the number of edges from v to u and from v to w. is always a vector representation of G v when v = 0. If, in addition, V is a By construction Pv V vector representation of minimal rank we obtain the following inequalities: K mr+ (G v) + 1
K rank Pv V + 1 = rank V = mr+ (G).
We record this as a proposition. Proposition 2.1 [4]. Let G to v by a single edge. Then K mr+ (G v) + 1
= (V , E ) be a multigraph with v ∈ V such that there exists w ∈ N (v) adjacent
K mr+ (G).
The existence of such a w is necessary, otherwise there exists minimal vector representations of G in which v is represented by the zero vector. We distinguish such vertices with the following definition. Definition 2.2. Let G = (V , E ) be a multigraph. A vertex v exists w ∈ N (v) adjacent to v by a single edge.
∈ V is called nondegenerate in G if there
Observation 2.3. Let G = (V , E ) be a multigraph. A vertex v ∈ V is nondegenerate in G if and only if v is represented by a nonzero vector in every vector representation of G.
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3. The reduction number In this section we describe an interpretation of the OS-number which was introduced in [15]. Observe that Proposition 2.1 implies a lower bound for the semidefinite minimum rank: for example if v1 is nondegenerate in G then K (G) mr+
K mr+ (G v1 ) + 1 1.
If in addition there is a vertex v2 which is nondegenerate in G v1 then we also have K mr+ (G)
K mr+ (G v1 v2 ) + 2 2.
In general, if there exists a list of vertices (v1 , . . . , vd ) in G such that v1 is nondegenerate in G and vi+1 K (G) d. is nondegenerate in G v1 · · · vi then Proposition 2.1 implies that mr+ With this motivation, the reduction number of a graph G, denoted R(G), is defined to be the following: R (G)
= max{R (G v) + 1 : v ∈ V (G), v nondegenerate in G}
where max ∅
= 0. The discussion above implies the following.
Theorem 3.1 [15, Theorem 4.1]. R (G)
K (G). mr+
If (v1 , . . . , vd ) is a list of vertices of G such that v1 is nondegenerate in G and vi+1 is nondegenerate in G v1 · · · vi , then we say that (v1 , . . . , vd ) is a reduction order of G. Clearly R (G) d if and only if there is a reduction order of length d. Mitchell et al. also show that the reduction number yields the same lower bound as the OS-number and positive semidefinite zero forcing number. Theorem 3.2 [15, Theorem 4.4]. Let G be a graph, then
|G| − Z+ (G) = OS(G) = R (G). Moreover every OS-set is a reduction order, while every reduction order can be permuted to yield an OS-set. Remark 3.3. Notice that the reduction number is defined for multigraphs. Originally the positive semidefinite zero forcing number and OS-number were only defined for simple graphs. However, Ekstrand et al. [6] extended the definition of Z+ (G) to multigraphs and Mitchell et al. [15] extended the definition of OS(G) to multigraphs. 4. Refining the reduction number for sign patterns To motivate the construction of R (G, f ) and R s (G) as described in the introduction, we consider the following example. Example 4.1. Suppose G is a connected simple graph with vertices u, v, w such that G[u, v, w] = K3 . be a vector representation of G and suppose u, v, w represent u, v, w. As in Section 2, let U = Let V Span{ v} and Pv : Rm → U ⊥ be the orthogonal projection onto the orthogonal complement of U. Then
Pv u, Pv w = u, w −
u, v v, w v, v
(3)
could be positive, zero, or negative. This uncertainty is modeled by the vertices u and w being adjacent is actually a vector representation of the signed graph by multiple edges in G v. Now suppose V (G, f ) and f ({u, v}) = f ({u, w}) = f ({v, w}) = −1. Then we know that the expression in Equation 3 is actually negative. The moral of Example 4.1 is that knowing the sign of each inner product in a vector representation can help prevent uncertainty in orthogonal removal. We now define sign patterns for multigraphs,
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construct a signed version of orthogonal vertex removal, and finally define a signed version of the reduction number. 4.1. Sign patterns of multigraphs If (G, f ) is a signed multigraph let PR (G, f ) denote the set of all psd matrices with entries in where:
R
• aij > 0 if every edge e connecting vertex i and vertex j has f (e) = 1, • aij < 0 if every edge e connecting vertex i and vertex j has f (e) = −1, • aij = 0 if vertex i and vertex j are not adjacent and i = j. No restriction is placed on aij if vertex i and vertex j are connected by edges e and e with f (e) The positive semidefinite minimum rank of a signed multigraph (G, f ) is defined as: K mr+ (G, f )
= f (e ).
= min{rank A : A ∈ PR (G, f )}.
Suppose that G is a multigraph, then the definitions above imply that PR (G)
= ∪ PR (G, f )
and so R mr+ (G, f )
R = min mr+ (G, f ). f
Remark 4.2. Let (G, f ) be a signed multigraph and suppose i, j ∈ V (G). Let e1 , . . . , em be the edges incident to i and j. The definition of PR (G, f ) has the following motivation. For A ∈ PR (G, f ) we think of the entry aij as being the sum of m numbers: aij
=
m
bk
k=1
where bk has the same sign as f (ek ). In particular, if each f (ek ) is negative then aij is negative but if some of the f (ek ) are positive and some are negative then aij could be positive, zero, or negative. 4.2. Signed orthogonal vertex removal We are now ready to construct the lower bound R (G, f ) for signed graphs (G, f ). Let G = (V , E ) be a graph with sign pattern f : E → {1, −1}. Let v ∈ V be a vertex, we wish to construct a sign pattern f v for the graph G v. The signed orthogonal removal of v from (G, f ) is the signed graph (G v, f v) where G v is the orthogonal removal of v from G described in Section 2 and f v is defined as follows: let N (v) = {w1 , . . . , wn } and let e1i , . . . , eni i be the edges connecting v to wi in G, then wi and wj have ni nj additional edges between them in G v. Label these edges ei,j,α,β with 1 α ni and 1 β nj . Finally define (f v)(e) = f (e) if e ∈ E (G) and j
(f v)(ei,j,α,β ) = −f (eαi )f (eβ ).
(4)
⊂ Rd is a vector representation of the signed graph (G, f ), we again construct the vector Now if V space U = Span{ v} and the orthogonal projection Pv : Rd → U ⊥ onto its orthogonal complement. is a vector representation of (G v, f v) when v = 0. If, in v = 0 element, Pv V Ignoring the Pv is a vector representation of minimal rank we obtain the following inequalities addition, V R mr+ (G v, f
R v) + 1 rank V = mr+ (G, f ).
We record this as a proposition.
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Proposition 4.3. Let (G, f ) be a signed graph and v v to w have the same sign. Then R mr+ (G v, f
∈ V (G) with w ∈ N (v) such that all edges connecting
R v) + 1 mr+ (G, f ).
The existence of such a w is necessary, otherwise there exists vector representations of (G, f ) in which v is represented by the zero vector. As before, we distinguish such vertices with the following definition. Definition 4.4. Let (G, f ) be a signed graph. A vertex v ∈ V is called nondegenerate in (G, f ) if there exists w ∈ N (v) such that every edge connecting v and w has the same sign. Observation 4.5. Let (G, f ) be a signed graph. A vertex v ∈ V is nondegenerate in (G, f ) if and only if v is represented by a nonzero vector in every vector representation of (G, f ). 4.3. The signed reduction number Now define the reduction number of a signed graph (G, f ) as R (G, f )
= max{R (G v, f v) + 1 : v nondegenerate in (G, f )}.
If (v1 , . . . , vd ) is a list of vertices of G such that v1 is nondegenerate in (G, f ) and vi+1 is nondegenerate in (G v1 · · · vi , f v1 · · · vi ), then we say that (v1 , . . . , vd ) is a reduction order of (G, f ). Proposition 4.6. Let (G, f ) be a signed graph then R (G, f )
R mr+ (G, f )
and
|G| − Z+ (G) = OS(G) = R (G) R (G, f ). Proof. The first claim follows from Proposition 4.3. We prove the second claim by induction on |G|. When |G| = 1 the claim is clearly true. Now assume |G| > 1. Either R (G) = 0 and we are done or there exists v ∈ V such that R (G v)+ 1 = R (G) and v is nondegenerate in G. Then v is nondegenerate in (G, f ) and so R (G)
= R (G v) + 1 R (G v, f v) + 1 R(G, f ).
Next, define the signed reduction number of a graph G to be R s (G)
= min R (G, f ). f
We then have the first main result of the paper. Theorem 4.7. For a graph G, |G| − Z+ (G)
R (G). = OS(G) = R (G) R s (G) mr+
Proof. The first inequality follows from Proposition 4.6 while the second follows from Equation 2.
5. A natural equivalence on the set of sign patterns of a fixed graph Assume (G, f ) is a signed graph and v ∈ G, then define a new sign pattern Tv f on G by ⎧ ⎨ −f (e) : e incident to v, Tv f (e) = ⎩ f (e) : otherwise.
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Next define an equivalence relation on the sign patterns of G as follows: f
f
= Tv1 Tv2 . . . Tvk g for some v1 , v2 , . . . , vk ∈ G.
∼ g if and only if
The next observation follows directly from the definition of orthogonal removal. Observation 5.1. Suppose (G, f ) is a signed graph with vertex v. Then (Tv f ) v is a vertex of G then Tw (f v) = (Tw f ) v.
= f v and if w = v
R and R depend only on the equivalence As the next proposition shows, the rank parameters mr+ class of f .
Proposition 5.2. Let G be a graph with sign patterns f and g. If f R (G, f ) = R (G, g ).
R (G, f ) = mrR (G, g ) and ∼ g then mr+ +
Proof. Notice that if f ∼ g then PR (G, f ) = D PR (G, g )D−1 where D is a diagonal matrix whose diagR (G, f ) = mrR (G, g ). onal entries consists of 1 and −1. As rank A = rank DAD−1 this implies that mr+ + If f ∼ g and v is a vertex in G, then Observation 5.1 implies that f v ∼ g v. This shows that (v1 , . . . , vd ) is a reduction order of (G, f ) if and only if it is a reduction order of (G, g ) and in particular R (G, f ) = R (G, g ). Proposition 5.3. Let G be a graph with sign pattern f . If T is a simple subtree of G then there exists a sign pattern g of G such that f ∼ g and g |E(T ) ≡ −1. Proof. Induct on |T |. If |T | = 1 then there is nothing to show. Otherwise T has a pendant vertex v. Let e be the unique edge in T incident to v. By induction if T = T − v then there exists a sign pattern g of G such that f ∼ g and g |E(T ) ≡ −1. Now let g = Tv (g ) or g = g depending on the sign of g (e). Proposition 5.3 allows us to pick “nice” representatives of each equivalence class. Example 5.4. Let G be a graph and T be a simple spanning tree of G. Then the set
f : E (G) → {+1, −1} : f |E(T ) ≡ −1 forms a “set of class representatives” of the equivalence ∼ (i.e. each equivalence class is represented exactly once). R (G , f ) 6. Computations for mr+ R (G, f ) for a general graph and a general sign pattern seems very difficult, but in Computing mr+
R (G, f ) for some special cases using the lower bound R (G, f ). this section we compute mr+ R (G, f ) Observation 6.1. For a signed graph (G, f ), mr+
Proof. If A
|G| − 1.
|G| − 1.
∈ PR (G, f ) has smallest eigenvalue λ0 then A − λ0 I|G| ∈ PR (G, f ) and rank(A − λ0 I|G| )
R (T , f ) Corollary 6.2. If T is a simple tree, then mr+
= R (T , f ) = |T | − 1 for any sign pattern f .
R (T , f ). Otherwise T has a pendant vertex v and (T v, f v) Proof. If |T | = 1 then R (T , f ) = 0 = mr+ is a signed simple tree. Then by induction R mr+ (T , f )
R (T , f ) R (T v, f v) + 1 = |T | − 1.
For a graph G = (V , E ) let nG : E → {1, −1} be the negative sign pattern, that is nG (e) ˇ [17] proved the following. all e ∈ E. In the language of vector representations, Šinajová
= −1 for
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Theorem 6.3 [17]. For any connected graph G and sign pattern f
R (G, f ) = R (G, f ) = |G| − 1. ∼ nG , mr+
Proof. The statement is clearly true if |G| = 1. If v ∈ V , then G v is a connected graph on |G| − 1 vertices and nG v is the negative sign pattern on G v. Thus by induction: R mr+ (G, nG )
R (G, nG ) R (G v, nG v) + 1 = |G| − 1.
Proposition 6.4. Let G = (V , E ) be a graph and f , g only one edge, then R R mr+ (G, f ) − mr+ (G, g ) 1.
= (v1 , . . . , vn ) Proof. Let V = g ({v1 , v2 }), then W
: E → {1, −1} be sign patterns of G that differ at
⊂ Rd be a vector representation of (G, f ) of minimum rank. If f ({v1 , v2 })
= (v1 ⊕ (α), v2 ⊕ (1), v3 ⊕ (0), . . . , vn ⊕ (0)) ⊂ Rd+1
R (G, g ) α is chosen correctly. This implies that mr+ R mr+ (G, f ) + 1. Repeating the argument starting with a vector representation of (G, g ) gives the other
is a vector representation of (G, g ) when inequality.
R (G) Theorem 6.5. If mr+
R (G, f ) = d. d |G| − 1 then there exists a sign pattern f such that mr+
R (G) = mrR (G, g ). Proof. By definition there exists a sign pattern g : E → {1, −1} such that mr+ + Construct a sequence of sign patterns g = g0 , g1 , . . . , gm = nG such that gi and gi+1 differ at exactly one edge. Then by Proposition 6.4 and Proposition 6.3 R R R R {mr+ (G, g0 ), . . . , mr+ (G, gm )} = {mr+ (G), mr+ (G) + 1, . . . , |G| − 1}.
1
For a signed simple cycle (Cn , f ) with vertices V (Cn ) = {1, . . . , n} and edges E (Cn ) = {{i, i + 1} : i n} we define the parameter: sign(Cn , f )
= (−1)n−1
n
i=1
f ({i, i + 1}).
Proposition 6.6. With the above notation, for n 3 ⎧ ⎨ n − 2 : if sign(C , f ) n R mr+ (Cn , f ) = R (Cn , f ) = ⎩ n − 1 : if sign(C , f ) n
= 1, = −1.
Proof. It is clear that if f ∼ g then sign(Cn , f ) = sign(Cn , g ). Further there are exactly two equivalence classes of sign patterns and sign(Cn , nCn ) = −1. The proposition then follows from Proposition 5.2, K (C ) Theorem 6.3, Theorem 6.5, and the fact that mr+ n
= n − 2.
7. Proof of Theorem 1.4 7.1. Graphs for which |G| − Z+ (G)
R (G) = R (G) = OS(G) < mr+
K (G) > R (G) come from the Möbius ladder on 8 vertices A major source of examples for which mr+ (see Fig. 1) which we denote ML8 . Mitchell et al. [14] showed that
5
K = mr+ (ML8 ) > R (ML8 ) = 4.
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K (G) by considering vertex sums Mitchell et al. constructed additional graphs for which R (G) < mr+ and joins of graphs. A vertex v in a connected graph G = (V , E ) is a cut-vertex if G[V − v] is disconnected. A maximal connected induced subgraph without a cut-vertex is called a block. Booth et al. [3] and van der Holst [18] have independently shown that the positive semidefinite minimum rank is completely determined by the blocks of G independently of how they are joined. More specifically: if G1 , ..., Gk are the blocks K (G) = k mrK (G ). The reduction number, like the positive semidefinite minimum of G then mr+ + i i=1 rank, behaves nicely over the blocks of G.
Theorem 7.1 [14, Theorem 4.1]. If G1 , ..., Gk are the blocks of a simple graph G then R (G)
=
k
i=1
R (Gi ).
A graph G is a vertex sum of two graphs G1 and G2 if G is the superposition of G1 and G2 identified at a vertex v. By letting G be a vertex sum of N copies of ML8 we see that 4N
K = R (G) < mr+ (G) = 5N
K (G) − R (G) can be arbitrarily large. showing that mr+ Given two graphs G1 and G2 the join of G1 and G2 , denoted G1 V (G1 ) ∪ V (G2 ) and edge set E (G1 ) ∪ E (G2 ) ∪ E where
E
∨ G2 , is the graph with vertex set
= {uv : u ∈ V (G1 ) and v ∈ V (G2 )}.
C (G ∨ G ) = max{mrC (G ∨ K ), mrC (G ∨ K )}. Barioli et Hackney et al. [10] showed that mr+ 1 2 1 1 + 1 + 2 al. [2] proved this in the real case. The reduction number also behaves like the positive semidefinite minimum rank when considering joins of graphs.
Theorem 7.2 [14, Theorem 4.3]. If G1 and G2 are connected simple graphs on two or more vertices, then R (G1
∨ G2 ) = max{R (G1 ), R (G2 )}.
Using Theorem 7.2, Mitchell et al. constructed for each N such that K mr+ (Gt ) − R (Gt )
> 0 an infinite number of graphs {Gt }t 2
= N.
For example one can set Gt = H ∨ Kt where H is a graph with exactly N blocks each identical to ML8 and Kt is the complete graph on t vertices. 7.2. Vertex sums and graph joins for the parameter R s (G) R (ML ) = 5 and state variants of Theorem 7.1 and 7.2 for We will next show that R s (ML8 ) = mr+ 8 the signed reduction number. This will prove Theorem 1.4. We begin with a lemma.
Lemma 7.3. Suppose (G, f ) is a signed graph. If H is an induced subgraph of G and h R (H , h) R (G, f ). In particular, R s (H ) R s (G).
= f |E(H ) then
Proof. Induct on R (H , h). If R (H , h) = 0 then the lemma follows. Now suppose R (H , h) > 0, then there exists a nondegenerate vertex v in (H , h) such that R (H , h) = R (H v, h v) + 1. As H is an induced subgraph of G, v is also nondegenerate in (G, f ). Further H v is an induced subgraph of G v and h v = (f v)|E(H v) so induction implies that R (H , h)
= R (H v, h v) + 1 R (G v, f v) + 1 R (G, f ).
Proposition 7.4. R s (ML8 )
= 5.
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Fig. 1. The graph ML8 and the graph ML8 with some distinguished edges.
Proof. Fix a sign pattern f of ML8 . We wish to show that R (ML8 , f ) replace f with an equivalent sign pattern (see Fig. 1) such that f (12)
5. Using Proposition 5.3 we can
= f (23) = f (26) = f (34) = f (37) = f (45) = f (48) = −1. = G[{1, 2, 3, 4, 5, 7}] and h = f |E(H ) then h ≡ −1.
Case 1: Suppose f (15) = −1. In this case, if H So Theorem 6.3 and Lemma 7.3 implies that R (G, f )
R (H , h) = |H | − 1 = 5.
Case 2: Suppose f (18) = −1 or f (56) = −1. Via the symmetry of ML8 we may suppose f (18) = −1. In this case, if H = G[{1, 2, 3, 4, 6, 8}] and h = f |E(H ) then h ≡ −1. So Theorem 6.3 and Lemma 7.3 implies that
R (H , h) = |H | − 1 = 5. Case 3: Suppose f (78) = −1 or f (67) = −1. Via the symmetry of ML8 we may suppose f (78) = −1. In this case, if H = G[{2, 3, 4, 5, 7, 8}] and h = f |E(H ) then h ≡ −1. So Theorem 6.3 and R (G, f )
Lemma 7.3 implies that
R (H , h) = |H | − 1 = 5. Case 4: Suppose f (15) = f (18) = f (56) = f (67) = f (78) = +1. In this case, if H = G[{1, 3, 4, 5, 6, 7}] and h = f |E(H ) then T1 T6 (h) ≡ −1. So Theorem 6.3 and Lemma 7.3 implies R (G, f )
that
R (G, f )
R (H , h) = |H | − 1 = 5.
Like the standard reduction number and OS-number, the signed reduction number is additive over the blocks of G. Theorem 7.5. If G1 , ..., Gk are the blocks of a simple graph G then R s (G)
=
k
i=1
R s (Gi ).
Theorem 7.5 follows from the next Lemma. Lemma 7.6. If v ∈ V and (G, f ) is simple and connected, then there exists a maximal signed reduction / S. order S of (G, f ) such that v ∈ The proof of Lemma 7.6 is somewhat involved so we delay it until the next section. Proof of Theorem 7.5. It is enough to show that R (G, f ) = R (G1 , f1 ) + R (G2 , f2 ) if G is a vertex sum of G1 and G2 at the vertex v and fi = f |E(Gi ) . Let S be a maximal reduction order of (G, f ) which does not contain the vertex v. Then Si = S ∩ V (Gi ) is a reduction order for (Gi , fi ) and so R (G, f )
= |S| = |S1 | + |S2 | R (G1 , f1 ) + R (G2 , f2 ).
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To see the other inequality, let Si be a maximal reduction order of (Gi , fi ) not containing the vertex v. Then S = S1 S2 is a reduction order of (G, f ) showing that R (G, f )
|S| = |S1 | + |S2 | = R (G1 , f1 ) + R (G2 , f2 ).
Corollary 7.7. The set Es is closed under vertex sums. Lemma 7.8. If G is simple and G
∈ Es then G ∨ v ∈ Es .
Proof. If G is disconnected and G1 , . . . , Gk are the connected components of G then v is a cut vertex R (G ∨ v) = mrR (G ∨ v) and R (G ∨ v) = R (G ∨ v). Hence we may of G ∨ v and so mr+ s s i + i assume that G is connected. If |G| = 1, then G ∨ v = K2 and the lemma is clearly true. Hence we may assume that G has two or more vertices. Then as G is connected and has two or more vertices R (G) = mrR (G ∨ v) [10, Corollary 2.7] and mr+ + R s (G)
R R = mr+ (G) = mr+ (G ∨ v) R s (G ∨ v) R s (G).
Notice that the last inequality follows from Lemma 7.3. Corollary 7.9. The set Es is closed under graph joins. Proof. Let G1 , G2 R (G1 mr+
Further as Gi
∈ Es then by Lemma 7.8,
R R ∨ G2 ) = max{mr+ (G1 ∨ v), mr+ (G2 ∨ v)} = max{R s (G1 ∨ v), R s (G2 ∨ v)}.
∨ v is an induced subgraph of G1 ∨ G2 , Lemma 7.3 implies that
max{R s (G1
R ∨ v), R s (G2 ∨ v)} R s (G1 ∨ G2 ) mr+ (G1 ∨ G2 ).
8. Proof of Lemma 7.6 The proof of Lemma 7.6 follows the proof of the corresponding result for OS-sets in [14, Proposition 2.16, Corollary 2.17], but the technical details in the signed case make the argument much more complicated. As in [14], we begin the proof of the lemma with two observations. Observation 8.1. If (v1 , . . . , vd ) is a reduction order of (G, f ) and vd is adjacent to wd in G v1 · · · vd−1 by edges of all the same sign, then (v1 , . . . , vd−1 , wd ) is a reduction order of (G, f ). Observation 8.2. Let (v1 , . . . , vd ) be a reduction order of (G, f ). If σ is a permutation of {1, . . . , d} such that σ (i) < σ (j) whenever i < j and vi and vj are in the same connected component of G[{v1 , . . . , vd }] then (vσ (1) , . . . , vσ (d) ) is a reduction order of (G, f ). Lemma 8.3. If (v1 , . . . , vd ) is list of vertices in (G, f ) and v is adjacent to w in (G, f ) v1 · · · vd by all edges of the same sign then v and w are either not adjacent or adjacent by all edges of the same sign in (G, f ) v1 · · · vβ−1 vβ+1 · · · vd . Proof. Let (G , f ) = (G, f ) v1 · · · vβ−1 and notice that (G , f ) − vβ vβ+1 · · · vd = (G , f ) vβ+1 · · · vd − vβ . In particular, it is enough to show that v and w are either not adjacent or adjacent by all edges of the same sign in (G , f ) − vβ vβ+1 · · · vd . This is easy to see: as (G , f ) − vβ is a sub sign pattern of (G , f ) vβ , the definition of orthogonal removal implies that (G , f ) − vβ vβ+1 · · · vd is a sub sign pattern of (G , f ) vβ vβ+1 · · · vd .
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Before stating the next lemma, we will need some notation. Notice that the map
(G, f ) → PR (G, f ) is not injective, many different signed graphs can determine the same set of positive semidefinite matrices. Because of this we will we introduce the concept of matrix equivalence between signed graphs. Consider the following two operations on a signed graph: (O1) Suppose (G, f ) is a signed graph and e is an edge in G incident to vertices x and y. Construct a new signed graph (G , f ) by adding an edge e to G incident to x and y such that f (e ) = f (e). e both incident to vertices x and y. If (O2) Suppose (G, f ) is a signed graph with edges e and f (e) = f (e ), construct a new signed graph G, f by removing the edge e from G. Definition 8.4. We say two signed graphs (G1 , f1 ) and (G2 , f2 ) are matrix equivalent and write (G1 , f1 ) ≈ (G2 , f2 ) if (G1 , f1 ) can be constructed from (G2 , f2 ) by repeated applications of operations (O1) and (O2) above. The motivation for the words “matrix equivalence” comes from the first part of the next observation. Observation 8.5. Suppose (G1 , f1 ) and (G2 , f2 ) are signed graphs. Then (1) (G1 , f1 ) ≈ (G2 , f2 ) if and only if PR (G1 , f1 ) = PR (G2 , f2 ), (2) If (G1 , f1 ) ≈ (G2 , f2 ) and v is a vertex in V (G1 ) = V (G2 ) then
(G1 , f1 ) v ≈ (G2 , f2 ) v. In general we cannot expect that G v u
= G u v, but we can prove the following.
Lemma 8.6. Suppose (G, f ) is a signed graph with vertices v and u. If u and v are either not adjacent or adjacent by edges of all the same sign in (G, f ), then
(G, f ) u v ≈ (G, f ) v u. Example 8.7. The assumption on u and v is necessary in Lemma 8.6. Take for instance the graph H1 in Fig. 2. Let e be the edge between u and v. Add another edge e between u and v and suppose f is a sign pattern such that f (e) = −1, f (e ) = 1, f (vx) = −1, and f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by one edge with sign −1. While in (G, f ) v u the vertices x and y are adjacent by five edges, two of these edges have sign +1 while three have sign −1. Hence, in this case, (G, f ) u v and (G, f ) v u are not matrix equivalent. Proof of Lemma 8.6. We begin by making a series of reductions: (1) if x, y are vertices of G and H = G[u, v, x, y] then (H , f ) v u = (G, f ) v u [x, y] and
(H , f ) u v = (G, f ) u v [x, y] hence we may assume |G| = 4 and V (G) = {x, y, u, v}, (2) we may assume u and v are adjacent in G, for otherwise it is clear that
(G, f ) v u = (G, f ) u v, (3) using reduction 1, we may assume {x, y} ⊂ N (v) ∪ N (u) for otherwise it is clear that (G, f ) v u = (G, f )[x, y] = (G, f ) u v,
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Fig. 2. The graphs H1 , H2 , H3 , and H4 .
(4) we may assume that x and y are not adjacent in G, for suppose e is an edge in G incident to x and y then ((G, f ) − e) v u = (G, f ) v u − e and ((G, f ) − e) u v = (G, f ) u v − e. So ((G, f ) − e) v u ≈ ((G, f ) − e) u v if and only if (G, f ) v u ≈ (G, f ) u v. With the reductions above and by possibly switching u and v or x and y, the underlying simple graph of G appears in Fig. 2. Now direct inspection shows that if two vertices in G are adjacent by edges of differing signs then x and y are adjacent by edges of differing signs in (G, f ) v u and (G, f ) u v. 1 Hence, in this case,
(G, f ) v u ≈ (G, f ) u v. So by replacing (G, f ) with a matrix equivalent signed graph we may suppose G is simple. Then using the graphs in Fig. 2 we have the following four cases. Case 1: G = H1 . By Observation 5.1, we may suppose that f (vu) = f (vx) = f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by one edge with sign −1. In (G, f ) v u, the vertices x and y are adjacent by two edges both with sign −1. Hence, in this case,
(G, f ) v u ≈ (G, f ) u v. Case 2: G = H2 . By Observation 5.1, we may suppose that f (yu) = f (uv) = f (vx) = −1. Then in both (G, f ) u v and (G, f ) v u, the vertices x and y are adjacent by one edge with sign −1. Hence, in this case,
(G, f ) v u = (G, f ) u v. Case 3: G = H3 . By Observation 5.1, we may suppose that f (vu) = f (vx) = f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by two edge with signs −1, f (ux). In (G, f ) v u, the vertices x and y are adjacent by three edges with signs −1, −1, f (ux). Hence, in this case,
(G, f ) v u ≈ (G, f ) u v. Case 4: G = H4 . By Observation 5.1, we may suppose that f (vu) = f (vx) = f (vy) = −1. Then in (G, f ) u v, the vertices x and y are adjacent by five edge with signs −1, f (ux), f (uy), −f (xu)f (uy), −f (xu)f (uy). In (G, f ) v u, the vertices x and y are adjacent by five edges with signs −1, −1, f (ux), f (uy), −f (xu)f (uy). Hence, in this case, (G, f ) v u ≈ (G, f ) u v. This completes the proof. Lemma 8.8. Let (v1 , . . . , vd ) be a reduction order of a signed graph (G, f ) such that vd is adjacent to wd in G v1 · · · vd−1 by all edges of the same sign. If C is a cycle in G[{v1 , . . . , vd } ∪ {wd }] containing the vertices vβ , vd , wd then
(v1 , . . . , vβ−1 , vβ+1 . . . , vd , wd ) and (v1 , . . . , vβ−1 , vβ+1 , . . . , vd , vβ ) are both reduction orders of (G, f ). 1 Notice that this assertion uses the fact that u and v are adjacent by all edges of the same sign, otherwise the last assertion would fail when G = H1 , see Example 8.7.
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Remark 8.9. When we say C is a cycle in H, we mean that C consists a list of distinct vertices c1 , . . . , cN in H and a list of edges e1 , . . . , eN such that ei is incident to ci and ci+1 (where cN +1 = c1 ). We do not assume C is an induced graph of H. Notice that a cycle on 2-vertices will be two vertices adjacent by / {c1 , . . . , cN } two edges. Also observe that if w ∈ H then there is a natural cycle C w in H w. If w ∈ we let C w = C. If w = ck for some k, then we let C w be the simple cycle with vertices c1 , . . . , ck−1 , ck+1 , . . . , cN and edges e1 , . . . , ek−2 , e , ek+1 , . . . , eN where e is the edge in H w coming from the edges ei−1 and ei in H (see Section 4.2). Proof of Lemma 8.8. Let
(Gk , fk ) = (G, f ) v1 · · · vβ+k and
(Hk , hk ) = (G, f ) v1 · · · vβ−1 vβ+1 · · · vβ+k . We will first prove the following claim. Claim: For 0 k d − β , (Hk , hk ) vβ ≈ (Gk , fk ). Induct on k. For k = 0, (H0 , h0 ) vβ claim holds for some k < d − β . Then
= (G0 , f0 ) and so the base case is true. Now suppose the
(Hk+1 , hk+1 ) vβ = (Hk , hk ) vβ+k+1 vβ and by Observation 8.5 and the induction hypothesis
(Gk+1 , fk+1 ) = (Gk , fk ) vβ+k+1 ≈ (Hk , hk ) vβ vβ+k+1 . In particular, if vβ and vβ+k+1 are not adjacent or adjacent by all edges of the same sign in Hk then Lemma 8.6 implies that
(Hk+1 , hk+1 ) vβ ≈ (Gk+1 , fk+1 ). Assume for a contradiction that vβ and vβ+k+1 are adjacent by edges of differing sign in Hk . By the hypothesis of the lemma and Remark 8.9 there exists a simple cycle C = vd wd c1 . . . c vβ c+1 . . . cN vd containing vd , wd , and vβ in Hk such that {c1 , . . . , cN } ⊂ {vk+β+1 , . . . , vd }.
/ {c1 , . . . , cN }. Then by the definition of orthogonal removal, c is adjacent Case 1: Suppose vβ+k+1 ∈ to vβ+k+1 by edges of differing signs in Hk vβ . The same is true for c+1 . Hence c and c+1 are adjacent to each other by edges of differing signs in Hk vβ vβ+k+1 . Now let σ be a permutation of {1, . . . , N } such that (cσ (1) , cσ (2) , . . . , cσ (N ) ) is sublist of (vk+β+2 , . . . , vd ). Then as (Hk , hk ) vβ ≈ (Gk , fk ), by Lemma 8.3 and Observation 8.5 the vertices vd and wd must either be nonadjacent or adjacent by all edges of the same sign in (Hk , hk ) vβ vβ+k+1 cσ1 · · · cσN . However, as c and c+1 are adjacent by edges of differing sign in (Hk , hk ) vβ vβ+k+1 and C vβ vβ+k+1 is a cycle containing c , c+1 , vd and wd this is impossible. So we have a contradiction and the proof of Claim 1 is complete in this case. Case 2: Suppose vβ+k+1 ∈ {c1 , . . . , cN }. Then by replacing C with a shorter cycle we may assume that vβ+k+1 is one of c or c+1 . Then by the definition of orthogonal removal, c and c+1 are adjacent to each other by edges of differing signs in Hk vβ . Now the same argument as above implies a contradiction. This completes the proof of the Claim. Now for 0
k d − β − 1 we have
(Gk , fk ) ≈ (Hk , hk ) vβ ,
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so vβ+k+1 is nondegenerate in (Hk , hk ) vβ and hence also nondegenerate in (Hk , hk ). Thus (v1 , . . . , vβ−1 , vβ+1 , . . . , vd−1 ) is a reduction order of (G, f ). So if
(H , h) = (G, f ) v1 · · · vβ−1 vβ+1 · · · vd−1 we must show that (vd , wd ) and (vd , vβ ) are both reduction orders in (H , h). Let (B, b) = (H , h) [vd , wd , vβ ], by the hypothesis of the lemma the underlying simple graph of B is the 3-cycle. Further, as vd and wd are adjacent by edges of the all the same sign in
(G, f ) v1 · · · vd = (Gd−β−1 , fd−β−1 ) ≈ (H , h) vβ we see that R (B, b) = 2. This implies, by Proposition 6.6, that b is equivalent to the all negative sign pattern. Then both (vd , wd ) and (vd , vβ ) are reduction orders of (B, b) and hence (H , h). Proof of Lemma 7.6. If there exists a maximal reduction order S of (G, f ) such that v ∈ / S we are done. So assume for a contradiction that every maximal reduction order of (G, f ) contains the vertex v. Then for a reduction order S = (u1 , . . . , ud ) of (G, f ) define the number NS to be the number of vertices uj picked after v such that uj and v are in the same connected component of G[S ]. If there exists a maximal reduction order S with NS = 0, then by Observation 8.2 we may rearrange our reduction order so that v is picked last. Then by Observation 8.1 there exists a reduction order S not containing v. Hence by our assumption, NS > 0 for every maximal reduction order. Let S = (u1 , . . . , ud ) be a reduction order such that NS is minimal. By Observation 8.2 we may assume that ud and v are in the same connected component of G[u1 , . . . , ud ]. Further, there exists a vertex wd adjacent to ud in G u1 · · · ud−1 by all edges of the same sign. We will construct a reduction order S such that NS < NS , thus contradicting the minimality of NS . Case 1: wd and v are in different connected component of G[{wd } ∪ {u1 , . . . , ud−1 }]. Then by Observation 8.1, S = (u1 , . . . , ud−1 , wd ) is a reduction order and NS < NS . Case 2: wd and v are in the same connected component of G[{wd } ∪ {u1 , . . . , ud−1 }]. Let H = G[{wd } ∪ {u1 , . . . , ud }]. By our assumptions, there exists a path P1 connecting v to wd in H that does not contain the vertex ud . As v and ud are in the same connected component of G[{u1 , . . . , ud }], there also exists a path P2 connecting v to ud in H that does not contain the vertex wd . Suppose v = uk and then using these paths we have the following two cases. Case 2(a): There is a simple cycle C in H containing wd , v, and ud . Then by Lemma 8.8, S = (u1 , . . . , uk−1 , uk+1 , . . . , ud , v) is a reduction order of (G, f ) and NS = 0. Case 2(b): There exists a simple cycle C in H containing wd , ud and a vertex uj such that wd and v are in different connected components of G[{wd } ∪ {u1 , . . . , ud } \ {uj }]. Then by Lemma 8.8, S = (u1 , . . . , uˆj , . . . , ud , wd ) is a reduction order of (G, f ) and NS < NS . Acknowledgements The author would like to thank Sivaram K. Narayan for his helpful comments on an earlier draft of this paper and the referee for pointing out an error in the original “proof” of Lemma 7.6. References [1] Francesco Barioli, Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Bryan Shader, P. van den Driessche, Hein van der Holst, Zero forcing parameters and minimum rank problems, Linear Algebra Appl. 433 (2) (2010) 401–411. [2] Francesco Barioli, Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Hein van der Holst, On the graph complement conjecture for minimum rank, Linear Algebra Appl. 436 (12) (2012) 4373–4391., Special Issue on Matrices Described by Patterns. [3] Matthew Booth, Philip Hackney, Benjamin Harris, Charles R. Johnson, Margaret Lay, Terry Lenker, Lon H. Mitchell, Sivaram K. Narayan, Amanda Pascoe, Brian D. Sutton, On the minimum semidefinite rank of a simple graph, Linear and Multilinear Algebra 59 (5) (2011) 483–506. [4] Matthew Booth, Philip Hackney, Benjamin Harris, Charles R. Johnson, Margaret Lay, Lon H. Mitchell, Sivaram K. Narayan, Amanda Pascoe, Kelly Steinmetz, Brian D. Sutton, Wendy Wang, On the minimum rank among positive semidefinite matrices with a given graph, SIAM J. Matrix Anal. Appl. 30 (2) (2008) 731–740.
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