Graph flattenability

Flattenability in some $$d$$-dimensional normed vector space is a property of graphs which states that any embedding, or drawing, of the graph in some high dimension $$d'$$ can be "flattened" down to live in $$d$$-dimensions, such that the distances between pairs of points connected by edges are preserved. A graph $$G$$ is $$d$$-flattenable if every distance constraint system (DCS) with $$G$$ as its constraint graph has a $$d$$-dimensional framework. Flattenability was first called realizability, but the name was changed to avoid confusion with a graph having some DCS with a $$d$$-dimensional framework.

Flattenability has connections to structural rigidity, tensegrities, Cayley configuration spaces, and a variant of the graph realization problem.

Definitions
A distance constraint system $$(G,\delta)$$, where $$G=(V,E)$$ is a graph and $$\delta: E \rightarrow \mathbb{R}^{|E|}$$ is an assignment of distances onto the edges of $$G$$, is $$d$$-flattenable in some normed vector space $$\mathbb{R}^d$$ if there exists a framework of $$(G,\delta)$$ in $$d$$-dimensions.

A graph $$G=(V,E)$$ is $$d$$-flattenable in $$\mathbb{R}^d$$ if every distance constraint system with $$G$$ as its constraint graph is $$d$$-flattenable.

Flattenability can also be defined in terms of Cayley configuration spaces; see connection to Cayley configuration spaces below.

Properties
Closure under subgraphs. Flattenability is closed under taking subgraphs. To see this, observe that for some graph $$G$$, all possible embeddings of a subgraph $$H$$ of $$G$$ are contained in the set of all embeddings of $$G$$.

Minor-closed. Flattenability is a minor-closed property by a similar argument as above.

Flattening dimension. The flattening dimension of a graph $$G$$ in some normed vector space is the lowest dimension $$d$$ such that $$G$$ is $$d$$-flattenable. The flattening dimension of a graph is closely related to its gram dimension. The following is an upper-bound on the flattening dimension of an arbitrary graph under the $$l_2$$-norm.

Theorem. The flattening dimension of a graph $$G = \left(V,E\right)$$ under the $$l_2$$-norm is at most $$O \left( \sqrt{\left| E \right|} \right)$$.

For a detailed treatment of this topic, see Chapter 11.2 of Deza & Laurent.

Euclidean flattenability
This section concerns flattenability results in Euclidean space, where distance is measured using the $$l_2$$ norm, also called the Euclidean norm.

1-flattenable graphs
The following theorem is folklore and shows that the only forbidden minor for 1-flattenability is the complete graph $$K_3$$.

Theorem. A graph is 1-flattenable if and only if it is a forest. Proof. A proof can be found in Belk & Connelly. For one direction, a forest is a collection of trees, and any distance constraint system whose graph is a tree can be realized in 1-dimension. For the other direction, if a graph $$G$$ is not a forest, then it has the complete graph $$K_3$$ as a subgraph. Consider the DCS that assigns the distance 1 to the edges of the $$K_4$$ subgraph and the distance 0 to all other edges. This DCS has a realization in 2-dimensions as the 1-skeleton of a triangle, but it has no realization in 1-dimension.

This proof allowed for distances on edges to be 0, but the argument holds even when this is not allowed. See Belk & Connelly for a detailed explanation.

2-flattenable graphs
The following theorem is folklore and shows that the only forbidden minor for 2-flattenability is the complete graph $$K_4$$.

Theorem. A graph is 2-flattenable if and only if it is a partial 2-tree.

Proof. A proof can be found in Belk & Connelly. For one direction, since flattenability is closed under taking subgraphs, it is sufficient to show that 2-trees are 2-flattenable. A 2-tree with $$n$$ vertices can be constructed recursively by taking a 2-tree with $$n-1$$ vertices and connecting a new vertex to the vertices of an existing edge. The base case is the $$K_3$$. Proceed by induction on the number of vertices $$n$$. When $$n=3$$, consider any distance assignment $$\delta$$ on the edges $$K_3$$. Note that if $$\delta$$ does not obey the triangle inequality, then this DCS does not have a realization in any dimension. Without loss of generality, place the first vertex $$v_1$$ at the origin and the second vertex $$v_2$$ along the x-axis such that $$\delta_{12}$$ is satisfied. The third vertex $$v_3$$ can be placed at an intersection of the circles with centers $$v_1$$ and $$v_2$$ and radii $$\delta_{13}$$ and $$\delta_{23}$$ respectively. This method of placement is called a ruler and compass construction. Hence, $$K_3$$ is 2-flattenable.

Now, assume a 2-tree with $$k$$ vertices is 2-flattenable. By definition, a 2-tree with $$k+1$$ vertices is a 2-tree with $$k$$ vertices, say $$T$$, and an additional vertex $$u$$ connected to the vertices of an existing edge in $$T$$. By the inductive hypothesis, $$T$$ is 2-flattenable. Finally, by a similar ruler and compass construction argument as in the base case, $$u$$ can be placed such that it lies in the plane. Thus, 2-trees are 2-flattenable by induction.

If a graph $$G$$ is not a partial 2-tree, then it contains $$K_4$$ as a minor. Assigning the distance of 1 to the edges of the $$K_4$$ minor and the distance of 0 to all other edges yields a DCS with a realization in 3-dimensions as the 1-skeleton of a tetrahedra. However, this DCS has no realization in 2-dimensions: when attempting to place the fourth vertex using a ruler and compass construction, the three circles defined by the fourth vertex do not all intersect.

Example. Consider the graph in figure 2. Adding the edge $$\bar{AC}$$ turns it into a 2-tree; hence, it is a partial 2-tree. Thus, it is 2-flattenable.

Example. The wheel graph $$W_5$$ contains $$K_4$$ as a minor. Thus, it is not 2-flattenable.

3-flattenable graphs
The class of 3-flattenable graphs strictly contains the class of partial 3-trees. More precisely, the forbidden minors for partial 3-trees are the complete graph $$K_5$$, the 1-skeleton of the octahedron $$K_{2,2,2}$$, $$V_8$$, and $$C_5 \times C_2$$, but $$V_8$$, and $$C_5 \times C_2$$ are 3-flattenable. These graphs are shown in Figure 3. Furthermore, the following theorem from Belk & Connelly shows that the only forbidden minors for 3-flattenability are $$K_5$$ and $$K_{2,2,2}$$. Theorem. A graph is 3-flattenable if and only if it does not have $$K_5$$ or $$K_{2,2,2}$$ as a minor.

Proof Idea: The proof given in Belk & Connelly assumes that $$V_8$$, and $$C_5 \times C_2$$ are 3-realizable. This is proven in the same article using mathematical tools from rigidity theory, specifically those concerning tensegrities. The complete graph $$K_5$$ is not 3-flattenable, and the same argument that shows $$K_4$$ is not 2-flattenable and $$K_3$$ is not 1-flattenable works here: assigning the distance 1 to the edges of $$K_5$$ yields a DCS with no realization in 3-dimensions. Figure 4 gives a visual proof that the graph $$K_{2,2,2}$$ is not 3-flattenable. Vertices 1, 2, and 3 form a degenerate triangle. For the edges between vertices 1- 5, edges $$(1,4)$$ and $$(3,4)$$ are assigned the distance $$\sqrt{2}$$ and all other edges are assigned the distance 1. Vertices 1- 5 have unique placements in 3-dimensions, up to congruence. Vertex 6 has 2 possible placements in 3-dimensions: 1 on each side of the plane $$\Pi$$ defined by vertices 1, 2, and 4. Hence, the edge $$(5,6)$$ has two distance values that can be realized in 3-dimensions. However, vertex 6 can revolve around the plane $$\Pi$$ in 4-dimensions while satisfying all constraints, so the edge $$(5,6)$$ has infinitely many distance values that can only be realized in 4-dimensions or higher. Thus, $$K_{2,2,2}$$ is not 3-flattenable. The fact that these graphs are not 3-flattenable proves that any graph with either $$K_5$$ or $$K_{2,2,2}$$ as a minor is not 3-flattenable. The other direction shows that if a graph $$G$$ does not have $$K_5$$ or $$K_{2,2,2}$$ as a minor, then $$G$$ can be constructed from partial 3-trees, $$V_8$$, and $$C_5 \times C_2$$ via 1-sums, 2-sums, and 3-sums. These graphs are all 3-flattenable and these operations preserve 3-flattenability, so $$G$$ is 3-flattenable.

The techniques in this proof yield the following result from Belk & Connelly.

Theorem. Every 3-realizable graph is a subgraph of a graph that can be obtained by a sequence of 1-sums, 2-sums, and 3-sums of the graphs $$K_4$$, $$V_8$$, and $$C_5 \times C_2$$.

Example. The previous theorem can be applied to show that the 1-skeleton of a cube is 3-flattenable. Start with the graph $$K_4$$, which is the 1-skeleton of a tetrahedron. On each face of the tetrahedron, perform a 3-sum with another $$K_4$$ graph, i.e. glue two tetrahedra together on their faces. The resulting graph contains the cube as a subgraph and is 3-flattenable.

In higher dimensions
Giving a forbidden minor characterization of $$d$$-flattenable graphs, for dimension $$d>3$$, is an open problem. For any dimension $$d$$, $$K_{d+2}$$ and the 1-skeleton of the $$d$$-dimensional analogue of an octahedron are forbidden minors for $$d$$-flattenability. It is conjectured that the number of forbidden minors for $$d$$-flattenability grows asymptotically to the number of forbidden minors for partial $$d$$-trees, and there are over $$75$$ forbidden minors for partial 4-trees.

An alternative characterization of $$d$$-flattenable graphs relates flattenability to Cayley configuration spaces. See the section on the connection to Cayley configuration spaces.

Connection to the graph realization problem
Given a distance constraint system and a dimension $$d$$, the graph realization problem asks for a $$d$$-dimensional framework of the DCS, if one exists. There are algorithms to realize $$d$$-flattenable graphs in $$d$$-dimensions, for $$d \leq 3$$, that run in polynomial time in the size of the graph. For $$d=1$$, realizing each tree in a forest in 1-dimension is trivial to accomplish in polynomial time. An algorithm for $$d=2$$ is mentioned in Belk & Connelly. For $$d=3$$, the algorithm in So & Ye obtains a framework $$r$$ of a DCS using semidefinite programming techniques and then utilizes the "folding" method described in Belk to transform $$r$$ into a 3-dimensional framework.

Flattenability under p-norms
This section concerns flattenability results for graphs under general $p$-norms, for $$1 \leq p \leq \infty$$.

Connection to algebraic geometry
Determining the flattenability of a graph under a general $$p$$-norm can be accomplished using methods in algebraic geometry, as suggested in Belk & Connelly. The question of whether a graph $$G=(V,E)$$ is $$d$$-flattenable is equivalent to determining if two semi-algebraic sets are equal. One algorithm to compare two semi-algebraic sets takes $$(4|E|)^{O\left(nd|V|^2\right)}$$ time.

Connection to Cayley configuration spaces
For general $$l_p$$-norms, there is a close relationship between flattenability and Cayley configuration spaces. The following theorem and its corollary are found in Sitharam & Willoughby.

Theorem. A graph $$G$$ is $$d$$-flattenable if and only if for every subgraph $$H=G \setminus F$$ of $$G$$ resulting from removing a set of edges $$F$$ from $$G$$ and any $$l^p_p$$-distance vector $$\delta_H$$ such that the DCS $$(H,\delta_H)$$ has a realization, the $$d$$-dimensional Cayley configuration space of $$(H,\delta_H)$$ over $$F$$ is convex.

Corollary. A graph $$G$$ is not $$d$$-flattenable if there exists some subgraph $$H=G \setminus F$$ of $$G$$ and some $$l^p_p$$-distance vector $$\delta$$ such that the $$d$$-dimensional Cayley configuration space of $$(H,\delta_H)$$ over $$F$$ is not convex.

2-Flattenability under the l1 and l∞ norms
The $$l_1$$ and $$l_{\infty}$$ norms are equivalent up to rotating axes in 2-dimensions, so 2-flattenability results for either norm hold for both. This section uses the $$l_1$$-norm. The complete graph $$K_4$$ is 2-flattenable under the $$l_1$$-norm and $$K_5$$ is 3-flattenable, but not 2-flattenable. These facts contribute to the following results on 2-flattenability under the $$l_1$$-norm found in Sitharam & Willoughby.

Observation. The set of 2-flattenable graphs under the $$l_1$$-norm (and $$l_{\infty}$$-norm) strictly contains the set of 2-flattenable graphs under the $$l_2$$-norm.

Theorem. A 2-sum of 2-flattenable graphs is 2-flattenable if and only if at most one graph has a $$K_4$$ minor.

The fact that $$K_4$$ is 2-flattenable but $$K_5$$ is not has implications on the forbidden minor characterization for 2-flattenable graphs under the $$l_1$$-norm. Specifically, the minors of $$K_5$$ could be forbidden minors for 2-flattenability. The following results explore these possibilities and give the complete set of forbidden minors.

Theorem. The banana graph, or $$K_5$$ with one edge removed, is not 2-flattenable.

Observation. The graph obtained by removing two edges that are incident to the same vertex from $$K_5$$ is 2-flattenable.

Observation. Connected graphs on 5 vertices with 7 edges are 2-flattenable.

The only minor of $$K_5$$ left is the wheel graph $$W_5$$, and the following result shows that this is one of the forbidden minors.

Theorem. A graph is 2-flattenable under the $$l_1$$- or $$l_{\infty}$$-norm if and only if it does not have either of the following graphs as minors:


 * the wheel graph $$W_5$$ or
 * the graph obtained by taking the 2-sum of two copies of $$K_4$$ and removing the shared edge.

Connection to structural rigidity
This section relates flattenability to concepts in structural (combinatorial) rigidity theory, such as the rigidity matroid. The following results concern the $$l^p_p$$-distance cone $$\Phi_{n,l_p}$$, i.e., the set of all $$l^p_p$$-distance vectors that can be realized as a configuration of $$n$$ points in some dimension. A proof that this set is a cone can be found in Ball. The $$d$$-stratum of this cone $$\Phi^d_{n,l_p}$$ are the vectors that can be realized as a configuration of $$n$$ points in $$d$$-dimensions. The projection of $$\Phi_{n,l_p}$$ or $$\Phi^d_{n,l_p}$$ onto the edges of a graph $$G$$ is the set of $$l^p_p$$ distance vectors that can be realized as the edge-lengths of some embedding of $$G$$.

A generic property of a graph $$G$$ is one that almost all frameworks of distance constraint systems, whose graph is $$G$$, have. A framework of a DCS $$(G,\delta)$$ under an $$l_p$$-norm is a generic framework (with respect to $$d$$-flattenability) if the following two conditions hold:


 * 1) there is an open neighborhood $$\Omega$$ of $$\delta$$ in the interior of the cone $$\Phi_{n,l_p}$$, and
 * 2) the framework is $$d$$-flattenable if and only if all frameworks in $$\Omega$$ are $$d$$-flattenable.

Condition (1) ensures that the neighborhood $$\Omega$$ has full rank. In other words, $$\Omega$$ has dimension equal to the flattening dimension of the complete graph $$K_n$$ under the $$l_p$$-norm. See Kitson for a more detailed discussion of generic framework for $$l_p$$-norms. The following results are found in Sitharam & Willoughby.

Theorem. A graph $$G$$ is $$d$$-flattenable if and only if every generic framework of $$G$$ is $$d$$-flattenable.

Theorem. $$d$$-flattenability is not a generic property of graphs, even for the $$l_2$$-norm.

Theorem. A generic $$d$$-flattenable framework of a graph $$G$$ exists if and only if $$G$$ is independent in the generic $$d$$-dimensional rigidity matroid.

Corollary. A graph $$G$$ is $$d$$-flattenable only if $$G$$ is independent in the $$d$$-dimensional rigidity matroid.

Theorem. For general $$l_p$$-norms, a graph $$G$$ is


 * 1) independent in the generic $$d$$-dimensional rigidity matroid if and only if the projection of the $$d$$-stratum $$\Phi^d_{n,l_p}$$ onto the edges of $$G$$ has dimension equal to the number of edges of $$G$$;
 * 2) maximally independent in the generic $$d$$-dimensional rigidity matroid if and only if projecting the $$d$$-stratum $$\Phi^d_{n,l_p}$$ onto the edges of $$G$$ preserves its dimension and this dimension is equal to the number of edges of $$G$$;
 * 3) rigid in $$d$$-dimensions if and only if projecting the $$d$$-stratum $$\Phi^d_{n,l_p}$$ onto the edges of $$G$$ preserves its dimension;
 * 4) not independent in the generic $$d$$-dimensional rigidity matroid if and only if the dimension of the projection of the $$d$$-stratum $$\Phi^d_{n,l_p}$$ onto the edges of $$G$$ is strictly less than the minimum of the dimension of $$\Phi^d_{n,l_p}$$ and the number of edges of $$G$$.