Unique Completability of Partially Filled Low Rank Positive - - PowerPoint PPT Presentation

Unique Completability of Partially Filled Low Rank Positive Semidefinite Matrices

Tibor Jord´ an (E¨

• tv¨
• s University, Budapest)

joint work with Bill Jackson (QM U. London) and Shin-ichi Tanigawa (CWI, Amsterdam and RIMS, Kyoto)

Graph Theory 2015, Nyborg, Denmark

Tibor Jord´ an Unique completability

Frameworks with semisimple underlying graphs

A graph G is called semisimple if it contains no parallel edges. (but may contain loops). A d-dimensional framework is a pair (G, p), where G = (V , E) is a semisimple graph and p : V → Rd is a map.

Tibor Jord´ an Unique completability

Frameworks with semisimple underlying graphs

A graph G is called semisimple if it contains no parallel edges. (but may contain loops). A d-dimensional framework is a pair (G, p), where G = (V , E) is a semisimple graph and p : V → Rd is a map.

Tibor Jord´ an Unique completability

Global completability

We say that two d-dimensional frameworks (G, q) and (G, p) are equivalent if pi, pj = qi, qj (for all ij ∈ E(G)) (1) and that they are congruent if (1) holds for every pair i, j in V (G) (including i, j with i = j). This is equivalent to saying that qi = Api for all i ∈ V for some fixed d × d orthogonal matrix A. We say that a d-dimensional framework (G, p) is globally (uniquely) completable in Rd if for every d-dimensional framework (G, q) which is equivalent to (G, p) we have that (G, q) and (G, p) are congruent.

Tibor Jord´ an Unique completability

Global completability

We say that two d-dimensional frameworks (G, q) and (G, p) are equivalent if pi, pj = qi, qj (for all ij ∈ E(G)) (1) and that they are congruent if (1) holds for every pair i, j in V (G) (including i, j with i = j). This is equivalent to saying that qi = Api for all i ∈ V for some fixed d × d orthogonal matrix A. We say that a d-dimensional framework (G, p) is globally (uniquely) completable in Rd if for every d-dimensional framework (G, q) which is equivalent to (G, p) we have that (G, q) and (G, p) are congruent.

Tibor Jord´ an Unique completability

Partially filled positive semidefinite matrices

Let M be an n × n positive semidefinite matrix of rank d. Then M = P⊤P for some d × n matrix P (that is, M is a Gram matrix). Hence it can be defined by specifying n points in Rd (corresponding to the columns pi, 1 ≤ i ≤ n of P). Thus the entry M[i, j] is equal to the scalar product pi, pj. Let M be a partially filled n × n positive semidefinite matrix of rank d. The given entries define an undirected graph G = (V , E)

• n V = {1, 2, ...n} in which two vertices i, j are adjacent if and
• nly if M[i, j] is given. The graph G and the columns of P give

rise to a d-dimensional framework.

Tibor Jord´ an Unique completability

Partially filled positive semidefinite matrices

Let M be an n × n positive semidefinite matrix of rank d. Then M = P⊤P for some d × n matrix P (that is, M is a Gram matrix). Hence it can be defined by specifying n points in Rd (corresponding to the columns pi, 1 ≤ i ≤ n of P). Thus the entry M[i, j] is equal to the scalar product pi, pj. Let M be a partially filled n × n positive semidefinite matrix of rank d. The given entries define an undirected graph G = (V , E)

• n V = {1, 2, ...n} in which two vertices i, j are adjacent if and
• nly if M[i, j] is given. The graph G and the columns of P give

rise to a d-dimensional framework.

Tibor Jord´ an Unique completability

The PSD matrix completion problem

Given a partially filled n × n matrix M and an integer d, is it completable to a PSD matrix of rank d? How can we determine a completion? is the completion unique? does unique completability depend only on the underlying semisimple graph, provided the set of entries of the d × n matrix P is generic? can we develop combinatorial algorithms (resp. sufficient conditions) for testing (resp. implying) unique completability?

Tibor Jord´ an Unique completability

Some combinatorial sufficient conditions

A preview of the combinatorial/graph theoretic results (given a partially filled n × n matrix M and an integer d): Suppose that at least ⌈n/2⌉ + 3 (resp. ⌈(n + d)/2⌉ + 2) entries are known in each row/column of the matrix. Then the rank-2 (resp. rank-d) PSD completion of M is unique. Suppose that there are at most n − d − 1 unknown entries. Then the rank-d PSD completion of M is unique.

Tibor Jord´ an Unique completability

Some combinatorial sufficient conditions

A preview of the combinatorial/graph theoretic results (given a partially filled n × n matrix M and an integer d): Suppose that at least ⌈n/2⌉ + 3 (resp. ⌈(n + d)/2⌉ + 2) entries are known in each row/column of the matrix. Then the rank-2 (resp. rank-d) PSD completion of M is unique. Suppose that there are at most n − d − 1 unknown entries. Then the rank-d PSD completion of M is unique.

Tibor Jord´ an Unique completability

Previous work on the PSD matrix completion problem

Testing completability is NP-hard in Rd for d ≥ 2 (M. E.-Nagy, M. Laurent, A. Varvitsiotis, 2013). So is testing global completability. Combinatorial approach inspired by rigidity theory (A. Singer and

• M. Cucuringu, 2010).

Connections and applications: Euclidean distance matrix completion (M. Laurent, A. Varvitsiotis, 2013), computer vision, machine learning, control.

Tibor Jord´ an Unique completability

Previous work on the PSD matrix completion problem

Testing completability is NP-hard in Rd for d ≥ 2 (M. E.-Nagy, M. Laurent, A. Varvitsiotis, 2013). So is testing global completability. Combinatorial approach inspired by rigidity theory (A. Singer and

• M. Cucuringu, 2010).

Connections and applications: Euclidean distance matrix completion (M. Laurent, A. Varvitsiotis, 2013), computer vision, machine learning, control.

Tibor Jord´ an Unique completability

Previous work on the PSD matrix completion problem

Testing completability is NP-hard in Rd for d ≥ 2 (M. E.-Nagy, M. Laurent, A. Varvitsiotis, 2013). So is testing global completability. Combinatorial approach inspired by rigidity theory (A. Singer and

• M. Cucuringu, 2010).

Connections and applications: Euclidean distance matrix completion (M. Laurent, A. Varvitsiotis, 2013), computer vision, machine learning, control.

Tibor Jord´ an Unique completability

Local completability

We say that (G, p) is locally completable in Rd if there exists an

• pen neighborhood N(p) of p in Rd|V | such that for any q ∈ N(p)

the equivalence of (G, q) to (G, p) implies that the two frameworks are congruent (or equivalently, if every continuous motion of the vertices of (G, p) in Rd which preserves equivalence must also preserve congruence).

Tibor Jord´ an Unique completability

Examples in R1

Tibor Jord´ an Unique completability

Examples in R1

Tibor Jord´ an Unique completability

Examples in R1

Tibor Jord´ an Unique completability

Examples in R1

Tibor Jord´ an Unique completability

Infinitesimal c-motions

A map ˙ p : V → Rd is called an infinitesimal c-motion of (G, p) if pi, ˙ pj + pj, ˙ pi = 0 (ij ∈ E) (2) The |E| × d|V |-matrix representing this system of linear equations with variables ˙ p is the completability matrix of (G, p), denoted by C(G, p). For example, if G is a graph with V (G) = {1, 2, 3, 4} and E(G) = {11, 12, 23, 24}, the completability matrix is:     1 2 3 4 11 2p1 12 p2 p1 23 p3 p2 24 p4 p2    .

Tibor Jord´ an Unique completability

Infinitesimal c-motions

A map ˙ p : V → Rd is called an infinitesimal c-motion of (G, p) if pi, ˙ pj + pj, ˙ pi = 0 (ij ∈ E) (2) The |E| × d|V |-matrix representing this system of linear equations with variables ˙ p is the completability matrix of (G, p), denoted by C(G, p). For example, if G is a graph with V (G) = {1, 2, 3, 4} and E(G) = {11, 12, 23, 24}, the completability matrix is:     1 2 3 4 11 2p1 12 p2 p1 23 p3 p2 24 p4 p2    .

Tibor Jord´ an Unique completability

Trivial c-motions

For any d × d skew-symmetric matrix S, the map ˙ p : V → Rd defined by ˙ pi = Spi for i ∈ V is an infinitesimal c-motion. The infinitesimal c-motions of this kind are called trivial. Therefore, if |V | ≥ d, then rankC(G, p) ≤ dn − d 2

• .

(3) The rank of C(G, p) is also bounded above by the number of edges in the complete semisimple graph on n vertices.

Tibor Jord´ an Unique completability

Infinitesimal completability

A framework (G, p) is said to be infinitesimally completable if rankC(G, p) = dn − d

2

• when n ≥ d or rankC(G, p) =

n+1

2

• when n ≤ d. It is c-independent if rankC(G, p) = |E|.

Tibor Jord´ an Unique completability

Bipartite frameworks

If G is bipartite with vertex bipartition {V1, V2}, then the map ˙ p defined by ˙ p(i) = Ap(i) for i ∈ V1 and ˙ p(j) = −ATp(j) for j ∈ V2 is also an infinitesimal c-motion of (G, p) for any d × d matrix A. Therefore, rank C(G, p) ≤ d|V | − d2 (4) if G is bipartite with |Vi| ≥ d, i = 1, 2.

Tibor Jord´ an Unique completability

Generic local completability

A map p : V → Rd is called generic if the set of coordinates of p is algebraically independent over the rational field. Thus the rank of C(G, p) will be the same for all generic realizations of G. One can show that infinitesimal completability is a sufficient condition for local completability, and that the two properties are equivalent when (G, p) is generic. We say that the graph G is locally completable or c-independent in Rd if some (or equivalently, every) generic realization of G in Rd is locally completable or c-independent.

Tibor Jord´ an Unique completability

Generic local completability

A map p : V → Rd is called generic if the set of coordinates of p is algebraically independent over the rational field. Thus the rank of C(G, p) will be the same for all generic realizations of G. One can show that infinitesimal completability is a sufficient condition for local completability, and that the two properties are equivalent when (G, p) is generic. We say that the graph G is locally completable or c-independent in Rd if some (or equivalently, every) generic realization of G in Rd is locally completable or c-independent.

Tibor Jord´ an Unique completability

Necessary conditions for independence in Rd

The upper bounds on the rank of the completability matrices of the subgraphs of G imply: Lemma Let G = (V , E) be c-independent in Rd. Then (i) iG(X) ≤ d|X| − d

2

• for all X ⊆ V with |X| ≥ d, and

(ii) for each bipartite subgraph H = (V1, V2; F) on vertex set X = V1 ∪ V2 with |Vi| ≥ d, i = 1, 2 we have iH(X) ≤ d|X| − d2. This pair of necessary conditions is sufficient in R1.

Tibor Jord´ an Unique completability

Necessary conditions for independence in Rd

The upper bounds on the rank of the completability matrices of the subgraphs of G imply: Lemma Let G = (V , E) be c-independent in Rd. Then (i) iG(X) ≤ d|X| − d

2

• for all X ⊆ V with |X| ≥ d, and

(ii) for each bipartite subgraph H = (V1, V2; F) on vertex set X = V1 ∪ V2 with |Vi| ≥ d, i = 1, 2 we have iH(X) ≤ d|X| − d2. This pair of necessary conditions is sufficient in R1.

Tibor Jord´ an Unique completability

Locally completable graphs on the line

Theorem (Singer and Cucuringu 2010) G is locally completable in R1 if and only if each connected component of G contains an odd cycle. Theorem (Singer and Cucuringu 2010) G is minimally locally completable in R1 if and only if each connected component of G contains exactly one cycle, which is

• dd.

Tibor Jord´ an Unique completability

A dependent graph (”non-rigid circuit”) in R2

Tibor Jord´ an Unique completability

Generic global completability

Theorem Global completability is not a generic property in R2. We say that a graph G is globally completable in Rd if every generic realization of G in Rd is globally completable.

Tibor Jord´ an Unique completability

Global completability example in R2

1 2 3 4 6 7 8 9 5

Tibor Jord´ an Unique completability

Globally completable graphs on the line

Theorem (Singer and Cucuringu 2010) G is globally completable in R1 if and only if G is connected and contains an odd cycle.

Tibor Jord´ an Unique completability

The rectangular model

One may also consider the unique completability of low rank rectangular matrices, i.e. rectangular matrices of the form P⊤Q for some d × n matrix P and d × m matrix Q. In this case the known entries of the rectangular matrix define a bipartite graph G = (V , E) with bipartition (U, W ) in which |U| = n, |W | = m, and an edge ij corresponds to the known scalar product of row i in P⊤ and column j in Q.

Tibor Jord´ an Unique completability

Bipartite frameworks

We say that two bipartite frameworks (G, p) and (G, q) are bicongruent if pi, pj = qi, qj holds for every pair i ∈ U and j ∈ W . The framework (G, p) is globally bicompletable if every framework which is equivalent to (G, p), is bicongruent to (G, p). Similarly, (G, p) is said to be locally bicompletable if there exists an open neighborhood N(p) of p such that for any q ∈ N(p) the equivalence of (G, q) to (G, p) implies that the two frameworks are bicongruent.

Tibor Jord´ an Unique completability

Infinitesimal bicompletability

As we noted above, we have rank C(G, p) ≤ d|V | − d2 (5) for all bipartite frameworks (G, p) with G = (U, V ; E), |U|, |V | ≥ d. The rank is also bounded above by |U| |W |. We say that (G, p) is infinitesimally bicompletable if rank C(G, p) = d|V | − d2 when min{|U| |W |} ≥ d and rank C(G, p) = |U| |V | when min{|U| |W |} < d.

Tibor Jord´ an Unique completability

Bicompletability

We say that a bipartite graph G is locally bicompletable in Rd if (G, p) is infinitesimally bicompletable for some (or equivalently, every) generic d-dimensional framework (G, p). We say that G is globally bicompletable in Rd if every generic d-dimensional framework (G, p) is globally bicompletable.

Tibor Jord´ an Unique completability

Bicompletability and completability

For a simple graph G we use G ◦ to denote the semisimple graph

• btained from G by adding a loop to each vertex. The complete

semisimple graph on n vertices (resp. on vertex set X) is denoted by K ◦)n (resp. K ◦(X)). Theorem Suppose that G = (U, W ; E) is a bipartite graph with |U|, |W | ≥ d and S = {u1, . . . , ud} is a set of d distinct vertices in U. Then (i) G is locally bicompletable in Rd if and only if G + = G ∪ K ◦(S) is locally completable in Rd, (ii) G is globally bicompletable in Rd if and only if G + = G ∪ K ◦(S) is globally completable in Rd.

Tibor Jord´ an Unique completability

Bicompletability and completability

For a simple graph G we use G ◦ to denote the semisimple graph

• btained from G by adding a loop to each vertex. The complete

semisimple graph on n vertices (resp. on vertex set X) is denoted by K ◦)n (resp. K ◦(X)). Theorem Suppose that G = (U, W ; E) is a bipartite graph with |U|, |W | ≥ d and S = {u1, . . . , ud} is a set of d distinct vertices in U. Then (i) G is locally bicompletable in Rd if and only if G + = G ∪ K ◦(S) is locally completable in Rd, (ii) G is globally bicompletable in Rd if and only if G + = G ∪ K ◦(S) is globally completable in Rd.

Tibor Jord´ an Unique completability

Coning

We use G ∗ {v} to denote the cone graph of G, that is, the graph

• btained by adding a new vertex v and connecting each vertex of

G to v by a new edge. For a framework (G, p), let p∗ be the extension of p to V (G) ∪ {v} by p∗(v) = 0. The following property was observed by Singer and Cucuringu. Proposition Let G = (V , E) be a simple graph and let (G, p) be a d-dimensional framework with p(v) = 0 for all v ∈ V . Then (G ◦, p) is infinitesimally (resp. globally) completable in Rd if and

• nly if (G ∗ {v}, p∗) is infinitesimally (resp. globally) rigid in Rd.

Tibor Jord´ an Unique completability

Coning

Tibor Jord´ an Unique completability

Whiteley showed that a graph G is rigid in Rd−1 if and only if G ∗ {v} is rigid in Rd. This fact and the above Proposition imply the following. Lemma Let G be a simple graph. Then G ◦ is locally completable in Rd if and only if G is rigid in Rd−1.

Tibor Jord´ an Unique completability

Completability and rigidity

Connelly and Whiteley showed that a graph G is globally rigid in Rd−1 if and only if G ∗ {v} is globally rigid in Rd. This fact and the above Proposition imply the following. Lemma Let G be a simple graph. Then G ◦ is globally completable in Rd if and only if G is globally rigid in Rd−1.

Tibor Jord´ an Unique completability

Cluster graphs

Let H = (V , E) be a loopless multigraph. The cluster graph induced by H, denoted by G ◦

H, is the graph obtained from H by

replacing each vertex v ∈ V by K ◦

d(v), and replacing each edge

st ∈ E by an edge between the clusters Cs and Ct in such a way that the edges of G ◦

H connecting distinct clusters are pairwise

• disjoint. (Cluster graphs correspond to body-and-bar frameworks in

rigidity theory.)

Tibor Jord´ an Unique completability

Cluster graph

Tibor Jord´ an Unique completability

A partially filled matrix with a cluster structure

Tibor Jord´ an Unique completability

Completability of cluster graphs

By using the characterization of rigid and globally rigid body-bar graphs in Rd we obtain: Theorem Let H be a multigraph. Then the cluster graph G ◦

H induced by H is

locally completable in Rd if and only if H contains d

2

• edge-disjoint spanning trees.

Theorem Let H be a multigraph. Then the cluster graph G ◦

H induced by H is

globally completable in Rd if and only if H − e contains d

2

• edge-disjoint spanning trees for all e ∈ E(H).

Tibor Jord´ an Unique completability

0-extension

Tibor Jord´ an Unique completability

Double 1-extension

Tibor Jord´ an Unique completability

Looped 1-extension

Tibor Jord´ an Unique completability

Extensions

Lemma Let G = (V , E) be a semisimple graph and G ′ = (V ′, E ′) be the graph obtained from G by a d-dimensional 0-extension (double 1-extension, looped 1-extension, resp.) operation. If G is c-independent (resp. locally completable) in Rd then G ′ is also c-independent (resp. locally completable) in Rd.

Tibor Jord´ an Unique completability

Local completability on the line II.

As noted earlier, the equivalence of (i) and (ii) was verified by Singer and Cucuringu, using a different approach. Theorem The following statements are equivalent for a graph G = (V , E): (i) G is minimally locally completable in R1; (ii) Each connected component of G contains exactly

• ne cycle, which is odd;

(iii) Each connected component of G can be constructed from a graph with one vertex with one loop by a sequence of one-dimensional 0-extensions and double-1-extensions.

Tibor Jord´ an Unique completability

Vertex-splitting

Tibor Jord´ an Unique completability

Vertex-splitting

The (d-dimensional) vertex-splitting operation (at vertex v1 with some fixed partition {U0, U∗, U1} of N(v1) with |U∗| = d) removes the edges between v1 and the vertices in U0, inserts a new vertex v0, and inserts new edges v0u for u ∈ U0 ∪ U∗. Lemma Let G = (V , E) be a graph and G ′ = (V ′, E ′) be the graph

• btained from G by a vertex-splitting at vertex v1. If G is

c-independent (resp. locally completable) in Rd then G ′ is also c-independent (resp. locally completable) in Rd.

Tibor Jord´ an Unique completability

Planar bipartite graphs

Theorem All planar bipartite graphs are c-independent in R2. Theorem A planar bipartite graph is locally bicompletable in R2 if and only if it is a quadrangulation.

Tibor Jord´ an Unique completability

Planar bipartite graphs

Theorem All planar bipartite graphs are c-independent in R2. Theorem A planar bipartite graph is locally bicompletable in R2 if and only if it is a quadrangulation.

Tibor Jord´ an Unique completability

Sparse planar graphs

A (2, 1)-sparse planar graph which is dependent in R2.

Tibor Jord´ an Unique completability

Completability stresses

Let G = (V , E) be a semisimple graph. We define the completability function fG : Rd|V | → R|E| by fG(p) =

• . . . , pu, pv, . . .
• (p ∈ Rd|V |).

Then fG is smooth. Notice also that the completion matrix C(G, p) is the Jacobian of fG at p. We say that ω : e ∈ E → ωe ∈ R is a completability stress of (G, p) if C(G, p)⊤ω = 0, that is, for each u ∈ V

• v∈N(u)

ωuvpv = 0. (6)

Tibor Jord´ an Unique completability

Completability stresses

Let G = (V , E) be a semisimple graph. We define the completability function fG : Rd|V | → R|E| by fG(p) =

• . . . , pu, pv, . . .
• (p ∈ Rd|V |).

Then fG is smooth. Notice also that the completion matrix C(G, p) is the Jacobian of fG at p. We say that ω : e ∈ E → ωe ∈ R is a completability stress of (G, p) if C(G, p)⊤ω = 0, that is, for each u ∈ V

• v∈N(u)

ωuvpv = 0. (6)

Tibor Jord´ an Unique completability

The stress matrix

The stress matrix associated to ω is the |V | × |V |-matrix Ω, where each column and each row are associated with a vertex in V and each entry is given by Ω[u, v] = ωuv. (7) The following was observed by Singer and Cucuringu. Proposition Let G = (V , E) be a graph and (G, p) be a d-dimensional framework such that p(V ) linearly spans Rd. Then for any completability stress ω : E → R of (G, p), P(p)Ω = 0. (8) In particular, the rank of Ω is at most n − d.

Tibor Jord´ an Unique completability

A sufficient condition for global completability

Singer and Cucuringu conjectured that having a maximum rank stress matrix implies global completability. Theorem Let G be a finite graph and let p : V → Rd be generic. Then G is globally completable if there is a completability stress ω of (G, p) with rank Ω = n − d. Theorem Let G be a globally completable graph in Rd, and let G ′ be a graph obtained from G by a simple 0-extension. Then G ′ is globally completable in Rd.

Tibor Jord´ an Unique completability

A sufficient condition for global completability

Singer and Cucuringu conjectured that having a maximum rank stress matrix implies global completability. Theorem Let G be a finite graph and let p : V → Rd be generic. Then G is globally completable if there is a completability stress ω of (G, p) with rank Ω = n − d. Theorem Let G be a globally completable graph in Rd, and let G ′ be a graph obtained from G by a simple 0-extension. Then G ′ is globally completable in Rd.

Tibor Jord´ an Unique completability

Sufficient conditions in terms of the minimum degree

Theorem Let G = (V , E) be a semisimple graph on n vertices. Suppose that δ(G) ≥ ⌈n/2⌉ + 2. Then G is locally completable in R2.

Tibor Jord´ an Unique completability

Vertex-redundant local completability

A graph is said to be vertex-redundantly locally completable if G − v is locally completable for all v ∈ V . Theorem Let G = (V , E) be a vertex-redundantly locally completable graph in Rd with |V | ≥ d + 1 for some d ≥ 2. Then G is globally completable in Rd.

Tibor Jord´ an Unique completability

Vertex-redundant local completability

A graph is said to be vertex-redundantly locally completable if G − v is locally completable for all v ∈ V . Theorem Let G = (V , E) be a vertex-redundantly locally completable graph in Rd with |V | ≥ d + 1 for some d ≥ 2. Then G is globally completable in Rd.

Tibor Jord´ an Unique completability

Sufficient conditions in terms of the minimum degree

Theorem Let G = (V , E) be a simple graph on n vertices. Suppose that δ(G) ≥ ⌈n/2⌉ + 3. Then G is globally completable in R2. Conjecture For every d ≥ 1 there is an integer cd such that every semisimple graph on n ≥ cd vertices with δ(G) ≥ (n + d)/2 is locally completable in Rd.

Tibor Jord´ an Unique completability

Sufficient conditions in terms of the minimum degree

Theorem Let G = (V , E) be a simple graph on n vertices. Suppose that δ(G) ≥ ⌈n/2⌉ + 3. Then G is globally completable in R2. Conjecture For every d ≥ 1 there is an integer cd such that every semisimple graph on n ≥ cd vertices with δ(G) ≥ (n + d)/2 is locally completable in Rd.

Tibor Jord´ an Unique completability

A weaker version of the conjecture

Theorem For all d ≥ 1 and ǫ > 0 there exists an integer N = Nd,ǫ such that every semisimple graph G on n > N vertices with δ(G) ≥ n(1 + ǫ)/2 is locally completable in Rd. Proof sketch: By the Erd˝

• s-Stone Theorem there exists an N such

that every semisimple graph G on n > N vertices with δ(G) ≥ n(1 + ǫ)/2 has a subgraph F isomorphic to Kd,d,d. It is not hard to show that F is a locally completable graph on 3d

• vertices. We can now choose a maximal locally completable

subgraph H of G (on t vertices, say) and show that (i) if t is less than n/2 then some vertex in G − H is adjacent to d vertices in H,

• r (ii) if t is at least n/2 then each vertex in G − H is adjacent to

d vertices in H. So H = G.

Tibor Jord´ an Unique completability

Sufficient conditions in terms of the number of (missing) edges

Theorem Let G = (V , E) be a simple graph on n ≥ 6d + 3 vertices. Suppose that G has at most n − d − 1 missing edges. Then G is globally completable in Rd. Proof sketch: the number of edges in G is at least n

2

• − (n − d − 1), which is larger than (1 −

1 2d+1)n2 2 since

n ≥ 3(2d + 1). Therefore by Tur´ an’s theorem G contains a subgraph H which is isomorphic to K2d+2. It is not hard to show that H is globally completable (not obvious). To conclude the proof we show that a spanning subgraph of G can be obtained from H by a sequence of simple 0-extensions.

Tibor Jord´ an Unique completability

The proof sketch (second half)

Let {v1, . . . , v2d+2} be the vertices of H, and consider an ordering {v1, v2, . . . , vn} of the vertices which starts with the vertices of H and satisfies d(vi, {v1, v2, . . . , vi−1}) ≥ d(vj, {v1, v2, . . . , vi−1}) (9) for all 2d + 3 ≤ i < j ≤ n. Such an ordering can be found greedily. We claim that for all 2d + 3 ≤ i ≤ n we have d(vi, {v1, v2, ..., vi−1}) ≥ d (which implies the statement of the theorem). Indeed, by assuming that the inequality fails for vi we can deduce that all vertices after vi send at most d − 1 edges back to the set {v1, v2, ..., vi−1}, which means that the number of missing edges is at least (n − i + 1)(i − d) ≥ n − d. This contradicts the fact that G has at most n − d − 1 missing edges.

Tibor Jord´ an Unique completability

Refined version in R2

Theorem (Berger, Kleinberg, Leighton 1999) Suppose that G has at most n − 5 missing edges. Then G can be

• btained from K5 by a sequence of degree-4 extensions.

Theorem Suppose that G has at most n − 3 missing edges. Then G can be

• btained from K5 − e or K6 minus a matching by a sequence of

0-extensions and edge additions. Theorem Let G = (V , E) be a simple graph on n ≥ 6 vertices. Suppose that G has at most n − 3 missing edges. Then G is locally completable in R2.

Tibor Jord´ an Unique completability

Refined version in R2

Theorem (Berger, Kleinberg, Leighton 1999) Suppose that G has at most n − 5 missing edges. Then G can be

• btained from K5 by a sequence of degree-4 extensions.

Theorem Suppose that G has at most n − 3 missing edges. Then G can be

• btained from K5 − e or K6 minus a matching by a sequence of

0-extensions and edge additions. Theorem Let G = (V , E) be a simple graph on n ≥ 6 vertices. Suppose that G has at most n − 3 missing edges. Then G is locally completable in R2.

Tibor Jord´ an Unique completability

Refined version in R2

Theorem (Berger, Kleinberg, Leighton 1999) Suppose that G has at most n − 5 missing edges. Then G can be

• btained from K5 by a sequence of degree-4 extensions.

Theorem Suppose that G has at most n − 3 missing edges. Then G can be

• btained from K5 − e or K6 minus a matching by a sequence of

0-extensions and edge additions. Theorem Let G = (V , E) be a simple graph on n ≥ 6 vertices. Suppose that G has at most n − 3 missing edges. Then G is locally completable in R2.

Tibor Jord´ an Unique completability

Semisimple graphs with missing edges

Theorem Let G = (V , E) be a semi-simple graph on n ≥ d vertices. Suppose that G has at most n − d missing edges. Then G is locally completable in Rd. The bound n − d is best possible. To see this consider the graph

• btained from a complete semisimple graph by attaching a vertex

v of degree d − 1.

Tibor Jord´ an Unique completability

Semisimple graphs

Theorem Let G = (V , E) be a semi-simple graph on n ≥ d vertices. Suppose that G has at most n − d − 1 missing edges. Then G is globally completable in Rd. The bound is best possible, which is illustrated by a graph

• btained from a complete semi-simple graph by attaching a vertex

v of degree d which has a loop on it.

Tibor Jord´ an Unique completability

Open questions

Characterize the locally (globally) completable graphs in Rd, for d ≥ 2. The configuration space of a framework (G, p) is the set of all q ∈ R|V |d such that (G, q) is equivalent to (G, p). It seems likely that if (G, p) is generic and globally completable, then (G − e, p) is either locally completable or has an unbounded configuration space for each e ∈ E. Hence it would be useful to determine when the configuration space of a (generic) framework in Rd is bounded. When does a given partially-filled Hermitian matrix have a unique complex completion? Note that, in this case, global completability is known to be a generic property by a result of Jackson and Owen.

Tibor Jord´ an Unique completability

Thank you.

An informal announcement ECCO XIXX (The 29th Conference of the European Chapter on Combinatorial Optimization) will be held in Budapest, Hungary on May 26 - 28, 2016. See http://ecco2016.weebly.com/. Plenary speakers: Andr´ as Frank (Hungary), David Pisinger (Denmark), Leo Liberti (France).

Tibor Jord´ an Unique completability

Thank you.

An informal announcement ECCO XIXX (The 29th Conference of the European Chapter on Combinatorial Optimization) will be held in Budapest, Hungary on May 26 - 28, 2016. See http://ecco2016.weebly.com/. Plenary speakers: Andr´ as Frank (Hungary), David Pisinger (Denmark), Leo Liberti (France).

Tibor Jord´ an Unique completability