Rooted-tree Decompositions with Matroid Constraints and the Infinitesimal Rigidity of Frameworks with Boundaries
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph , a multiset of vertices in , and a matroid on . We prove a necessary and sufficient condition for to be decomposed into edge-disjoint subgraphs such that (i) for each , is a tree with , and (ii) for each , the multiset is a base of . If is a free matroid, this is a decomposition into edge-disjoint spanning trees; thus, our result is a proper extension of Nash-Williams’ tree-partition theorem.
Such a matroid constraint is motivated by combinatorial rigidity theory. As a direct application of our decomposition theorem, we present characterizations of the infinitesimal rigidity of frameworks with non-generic “boundary”, which extend classical Laman’s theorem for generic 2-rigidity of bar-joint frameworks and Tay’s theorem for generic -rigidity of body-bar frameworks.
In this paper two fundamental results in combinatorial optimization, Tutte-Nash-Williams tree-packing theorem and Nash-Williams tree-partition theorem, are extended. In 1961 Tutte  and Nash-Williams  independently proved that an undirected graph contains edge-disjoint spanning trees if and only if holds for any partition of , where denotes the set of edges of connecting two distinct subsets of and denotes the number of subsets of . As a dual form, Nash-Williams tree-partition theorem  asserts that an undirected graph can be decomposed into edge-disjoint spanning trees if and only if and for any non-empty , where denotes the set of vertices incident to .
These two theorems are sometimes referred to in terms of rooted-edge-connectivity, as edge-disjoint spanning trees indicate how to send distinct “commodities” from a specific root-node to other vertices without interference. (In fact, the packing of spanning trees is an equivalent concept to rooted-edge-connectivity, see e.g., .) In this paper we address a more general situation. Suppose we have distinct roots, each of which has an ability of sending a commodity, and suppose the set of commodities possesses an independence structure, say, linear independence by regarding commodities as vectors. Then we are asked to decide whether one can send commodities from roots to every vertex so that each vertex receives independent commodities without transmitting more than two distinct commodities through an edge. This paper provides a polynomial time algorithm to answer to this question.
The study is motivated by combinatorial rigidity theory. One of major topics in rigidity theory is to describe a rigidity condition of architectural frameworks in terms of the underlying graphs, where the connection to tree-packing condition (and its variants) has been particularly investigated in the literature (see e.g.,[42, 35, 40]). Based on this background together with our new decomposition theorem, we obtain extensions of two fundamental theorems in combinatorial rigidity theory, Laman’s theorem for generic 2-rigidity of bar-joint frameworks and Tay’s theorem for generic -rigidity of body-bar frameworks.
1.1 Rooted-tree Decompositions
For a graph , a pair of and is called a rooted-tree if either (i) or (ii) is connected without cycles and . Here is called a root of . For a rooted-tree , we denote the set by , and we say that is spanned by if . Note that if ; otherwise (which is not equal to ).
As we mentioned, our focus is on a decomposition of a graph into edge-disjoint rooted-trees of specific roots. For simplicity, a pair of a graph and a multiset of vertices (that specify roots) is called a graph with roots.
Let be a graph with roots and be a matroid on . Rooted-trees are called edge-disjoint if for ; they are said to be basic if the multiset is a base of for each . We say that admits a basic rooted-tree decomposition with respect to (or simply, a basic decomposition) if the edge set can be partitioned into basic edge-disjoint rooted-trees , (where is allowed).
Figure 1 shows an example for the case when is a graphic matroid.
For each and , let and as multi-subsets of . The following main theorem characterizes the decomposability into basic edge-disjoint rooted-trees.
Let be a graph with a multiset of vertices, and be a matroid on of rank and the rank function . Then, admits a basic rooted-tree decomposition with respect to if and only if satisfies the following three conditions:
is independent in for each ;
for any non-empty ;
Notice that, if is a free matroid, this coincides with Nash-Williams’ tree-partition theorem. In Theorem 5.1, we give a dual form of Theorem 1.2 as a proper extension of Tutte-Nash-Williams’ tree-packing theorem.
Throughout the paper, we will refer to the conditions given in Theorem 1.2 as (C1), (C2) and (C3) with respect to , respectively. Checking (C2) can be easily reduced to a submodular function minimization and thus done in polynomial time. In Section 4 we present an efficient algorithm via matroid intersection.
Note that, even though checking (C2) can be reduced to matroid intersection, this fact alone does not imply Theorem 1.2. Indeed, if can be written as the direct sum of matroids of rank , Theorem 1.2 straightforwardly follows from the matroid union theorem; however for general Theorem 1.2 has no clear (and direct) connection to the matroid union theorem.
1.2 Related Works
Nash-Williams’ tree-partition theorem is nowadays a special case of the matroid union theorem, as it is equivalent to packing bases of the graphic matroid of (see e.g.,[29, 9]). For applications to rigidity theory, Whiteley  discussed a generalization of Nash-Williams’ theorem by mixing spanning trees and spanning pseudoforests. (A graph is said to be a spanning pseudoforest if each connected component contains exactly one cycle). Based on the matroid union theorem, he observed that, for two integers and with , can be partitioned into edge-disjoint spanning trees and spanning pseudoforests if and only if and for any non-empty . The range of was later broadened by Haas . Algorithms for checking these counting conditions or computing decompositions were discussed in e.g. [16, 33, 13, 3, 23, 17].
These types of matroids are referred to as count matroids  or sparsity matroids, and have a wide range of applications in combinatorial geometry, including rigidity theory (see, e.g.,). Our primary motivation of this study is indeed to extend the decomposition theory of these count matroids to more general forms. For this purpose, we have presented a special case of Theorem 1.2 in  where is restricted to a variant of uniform matroid.
Another direction of related research is the packing of branchings into digraphs. A directed forest, called a branching, is a digraph in which the in-degree of each node is at most one. The set of nodes of in-degree is called the root-set. For , a branching is said to be a spanning branching with roots if every vertex can reach to a root in . The well-known Edmonds branching-theorem  is a good characterization of a digraph with a given collection of root-sets to contain arc-disjoint spanning branchings with roots . However, Edmonds’ branching theorem can produce only spanning branchings; in general, the problem of answering whether there exist arc-disjoint branchings spanning a proper subset of is known to be NP-complete, and only a few special cases are known to be solvable in polynomial time [2, 19, 12].
Even in the undirected case, the problem becomes intractable if we drop the term “spanning” from the decomposition. In fact, the problem of deciding whether an undirected graph can be partitioned into two edge-disjoint trees is known to be NP-complete . Our main theorem (Theorem 1.2) however asserts that one can actually relax the condition of “spanning” by introducing an appropriate matroid constraint.
1.3 Applications to Rigidity Theory
Theorem 1.2 has various applications to rigidity theory. A bar-joint framework is a structure consisting of bars connected by universal joints at endpoints as shown in Figure 2(a). The underlying graph is obtained by associating each joint with a vertex and each bar with an edge, thus a bar-joint framework can be identified with a pair of a graph and . Celebrated Laman’s theorem  asserts that is minimally rigid on a generic in the plane if and only if and for any nonempty , where is called generic if the set of coordinates is algebraically independent over . See, e.g.,  for formal definition.
Although characterizing generic 3-dimensional rigidity of bar-joint frameworks is recognized as one of the most difficult open problems in this field, there are solvable structural models even in higher dimension. One of the fundamental results in this direction is a combinatorial characterization of generic rigidity of body-bar frameworks shown by Tay . Body-bar frameworks consist of disjoint rigid bodies articulated by bars as illustrated in Figure 3(a), and the underlying graphs are extracted by associating each body with a vertex and each bar with an edge. Tay  proved that the generic rigidity of body-bar frameworks can be characterized in terms of the underlying graphs by Nash-Williams’ condition for decomposing into spanning trees.
In this paper, replacing Nash-Williams’ theorem with Theorem 1.2, we obtain extensions of Laman’s theorem and Tay’s theorem to the models with boundary. In most applications, especially in engineering context, a framework has a relation to the external environment, where several joints/bodies are connected to the ground or walls. Figure 2(b) and Figure 3(b)(c) show typical examples: Figure 2(b) illustrates a so-called pinned bar-joint framework, where three joints are fixed in the space; in Figure 3(b) and (c) illustrate body-bar counterparts, where several bodies are linked to the ground by bars or pins. This motivates us to investigate frameworks with boundary.
Frameworks with boundary are indeed an old concept even in the mathematical study of rigidity (see  for survey and fundamental facts). In fact, combinatorial characterizations of these models straightforwardly follow from Laman’s theorem or Tay’s theorem, if we assume “genericity” of configuration of boundary. For example, to extend Laman’s theorem to pinned bar-joint frameworks, we just need to observe that a 2-dimensional pinned bar-joint framework is rigid if and only if there are at least two pinned joints and connecting all pairs of pinned joints results in a rigid framework (without pinning). This fact combined with Laman’s theorem implies a combinatorial characterization of 2-dimensional pinned bar-joint frameworks for generic rigidity. This straightforward extension however requires that should be generic and in particular pinned joints have to be generic, which cannot be achieved in most applications as joints are usually pinned down on the ground or walls.
Motivated by these practical requirements, we shall address the problem of coping with “non-generic” boundaries. Our new results assert that, even without genericity assumption for boundary condition, a naturally extended statement is true for characterizing infinitesimal rigidity. Although the formal description will be given in Sections 6 and 7, counting conditions (C1)(C2)(C3) of Theorem 1.2 will naturally appear as a necessary condition for the infinitesimal rigidity of frameworks with “non-generic” boundary, and the existence of basic rooted-tree decompositions enables us to show even the sufficiency.
Below, we list structural models we address in this paper:
bar-joint frameworks with bar-boundary in , in which the Plücker coordinate of each boundary-bar is predetermined (Theorem 7.3);
bar-joint frameworks with pin-boundary in , in which the coordinate of each pin is predetermined (Theorem 7.5);
bar-joint frameworks with slider-boundary in , in which the direction of each slider is predetermined (Theorem 7.6);
body-bar frameworks with bar-boundary in , in which the Plücker coordinate of each boundary-bar is predetermined (Theorem 6.1);
body-bar frameworks with pin-boundary in , in which the coordinate of each pin is predetermined (Theorem 6.3).
The second one (Theorem 7.5) was recently observed by Servatius, Shai and Whiteley  for engineering applications, where the proof is done by the the so-called Henneberg construction. We shall present it as a corollary of a more general statement (Theorem 7.3). We should note that main results of [30, 31] are a combinatorial characterization of assur graphs and their geometric properties in the plane. Our new observations for body-bar frameworks might be useful for developing a higher dimensional counterpart.
2-dimensional bar-joint frameworks with slider-boundary (called bar-joint-slider frameworks) were previously studied in Streinu and Theran , where an interesting relation between decompositions and non-generic realizations was observed. Theorem 7.6, which is a corollary of Theorem 7.5, extends their result. (This result was already presented in a conference  without detailed proof.)
We first review a combinatorial background in Section 2 and then present a proof of Theorem 1.2 in Section 3. In Section 4 we discuss computational issues. In Section 5 we present a dual form of Theorem 1.2. Applications of basic-decompositions to rigidity theory are discussed in Sections 6 and 7. We conclude the paper by listing remarks.
For a matroid on a finite set , the rank function of is denoted by . is especially called the rank of , which is simply denoted by . A set is called a spanning set of if . For , let . The restriction of to is , which forms a matroid on . The truncation of is defined as the one of rank function . An element is called a coloop if . is said to be parallel to if .
We will use the following preliminary result concerning the matroid induced by a monotone submodular function, which can be found in e.g. [27, Chapter 12]. The function is called submodular if for any and monotone if for any . Also is called intersecting submodular if the submodular inequality holds for every pair with .
Let be an integer-valued monotone submodular function. It is known that induces a matroid on , denoted by , whose collection of independent sets is written by . The following proposition provides an explicit formula expressing the rank function of , see e.g., [29, 11, 9].
Let be an integer-valued monotone submodular function on satisfying for every non-empty . Then, for any non-empty , the rank of in is given by
where the minimum is taken over all partitions of such that for each (and may be empty).
3 Proof of Theorem 1.2
Let be a graph with roots , be a matroid on with rank and the rank function . We begin with an easier direction, the necessity of Theorem 1.2.
Proof of the necessity of Theorem 1.2.
For a basic decomposition, (C1) is obviously necessary.
To see (C2) and (C3), let us take a basic rooted-tree decomposition of with respect to , where . can be converted to an arborescence (i.e., a directed tree) by assigning an orientation so that each vertex in has exactly one entering arc (and has no entering arc). Since the decomposition is basic, the sum of and the number of edges entering to is equal to for each . This implies , and thus (C3) holds.
To see (C2), let us consider , and let be the set of edges oriented from a vertex in to a vertex in . For the same reason as above, we have . Moreover, since the decomposition is basic, holds. These imply . ∎
For an integer , we define a set function by
Let be a graph with roots, be a matroid on , and be an integer. Suppose (C1) is satisfied and . Then, is an integer-valued monotone submodular function.
It is known that, for any , the set function defined by is monotone and submodular (see e.g., ). We now have . Since by (C1) and , is monotone and submodular. ∎
Thus, if (C1) is satisfied and , induces a matroid on , which is denoted by . Note that satisfies (C2) with respect to if and only if is independent in .
To show the sufficiency, we begin with an easy observation. is called disconnected if is not connected. A connected component is a subgraph of , where is a connected component of and .
Let be a disconnected graph with roots, and be a matroid on of rank . Suppose (C1), (C2) and (C3) are satisfied. Then, for each connected component of , is a spanning set of , and satisfies (C1), (C2) and (C3) with respect to .
Let be a connected component. Clearly, satisfies (C1).
From (C2) of , we have and . From (C3), . Also, for since is the rank of . Combining these relations, we have . In other words, the equality holds in each inequality, and in particular we have and . This implies the first part of the claim. Note . This implies (C3) of . Also, for any , we have , implying (C2) of . ∎
Let us move to the proof of the sufficiency of our main theorem.
Proof of the sufficiency of Theorem 1.2.
The proof is done by induction on . Note that, if , Theorem 1.2 trivially follows from (C1) and (C3), and hence we shall consider the case . If is disconnected, we can consider each connected component separately by Lemma 3.2. We thus assume that is connected.
For , denotes the subgraph edge-induced by . Namely, . A non-empty is said to be tight if . A tight set is called proper if . We begin with investigating properties of proper tight sets.
Suppose has a proper tight set . Let . Then there is an satisfying the following two properties:
admits a basic rooted-tree decomposition with respect to ,
can be partitioned into edge-disjoint spanning trees on .
Take a vertex . By (C1), we have . We insert copies of into as new roots (if ), and let be the resulting multiset. A new matroid on is constructed based on by adding these copies as coloops. Namely, . We now show
Clearly (C1) is satisfied. Since each element of is inserted as a coloop in , we have and thus . By the independence of in , is also independent in , implying (C2). Furthermore, since , (C3) is satisfied.
Thus, by induction on the size of edge set, admits a basic rooted-tree decomposition , where . Without loss of generality, let be the rooted-trees among them whose roots belong to . Since is a base of for each , every vertex of must be spanned by for all . Let . Then has the desired property. ∎
A tight set is called unbalanced if there is a vertex satisfying ; Otherwise is called balanced. A proper tight set given in Figure 4 is an example of unbalanced one. We now consider the case where has an unbalanced proper tight set.
Suppose has an unbalanced proper tight set . Then, admits a basic rooted-tree decomposition.
Let and . Without loss of generality, we denote . By Claim 3.3, can be partitioned into such that are basic edge-disjoint rooted-trees with respect to and is the union of edge-disjoint spanning trees on . (See Figure 4 for an example.) Then, we have
otherwise, for all ; as the decomposition is basic with respect to , we have ; thus, becomes balanced, a contradiction.
Based on , we now construct a new graph with roots in the following way:
Remove from and remove from ;
For each and for each , insert a copy of into as a new root. This copy is denoted by .
In total we inserted copies of each into as new roots, since there are exactly rooted-trees among that span . An example is given in Figure 5(a). We denote the multiset of these new roots by (i.e., ). We have thus constructed a new graph with and . From (4) and the construction, we have
A new matroid on is constructed from as follows. For each with and for each , we insert into so that is parallel to (in the sense of matroids). We then obtained a matroid on the multiset . After removing all elements of , a matroid, , on is defined. (See Figure 5(b).) From the construction we have, for each ,
We now claim the following:
|satisfies (C1) (C2) (C3) with respect to .||(8)|
Assuming (8) for a while, let us show how to construct a basic decomposition of . By (5), holds, and hence we can apply the inductive hypothesis to . Namely, admits a basic rooted-tree decomposition by induction (see Figure 5(c)). Recall that consists of and . It is thus convenient to denote the corresponding rooted-trees of the decomposition by and . (So is a partition of into edge-disjoint trees.) Note that, for any and any , holds by (7). This implies that cannot span from the basicness; in other words,
We are now ready to construct a basic rooted-tree decomposition of with respect to . For each , we define by
Clearly, is connected with (if ). By (9), has no cycle, and thus is a rooted-tree. Also, it is not difficult to see that each vertex is spanned by rooted-trees since there are exactly indices “” for which or span . We now check that this decomposition is indeed basic.
Consider , and suppose for some . From the construction of there is such that ; hence we obtain from definition (10). Namely, implies . Since is parallel to in , this implies
for each . Also, for each ,
from definition (10). We thus obtain, for each ,
Since each vertex is spanned by rooted-trees among , (11) implies that is basic. We thus obtained a basic rooted-tree decomposition of .
To see (C2), suppose for a contradiction that there is with that violates (C2). Namely, . Since satisfies , we must have
Also, since and is the union of edge-disjoint spanning trees with , can be partitioned into edge-disjoint forests, which implies