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3-Cycle-Free Persistence for Cooperative Control of Formations with Acyclic Control Structure
3-Cycle-Free Persistence for Cooperative Control of Formations with Acyclic Control Structure ? Tolga Eren Kirikkale University, Department of Electrical and Electronics Engineering, 71450 Kirikkale, Turkey (e-mail: [email protected]). Abstract: In this paper, we analyze the problem of acquiring 3-cycle-free persistent formations of mobile autonomous agents that have acyclic control structure. Nyquist-like criterion for formation stabilization is used in literature for the problem of relative formation stabilization. For this purpose, spectral properties of the Laplacian matrix are used in evaluating desirable structural properties of formations. Establishing measures of near-periodicity are useful in quantifying formation stability margins. In this paper, we analyze constructions to create non-3periodic persistent formations, i.e., we study the constructions to create 3-cycle-free persistent formations. Central to the development of our analysis will be the use of tools from rigidity theory and graph Laplacians. Keywords: Cooperative control systems, multi-agent systems, coordination of multiple vehicle systems, networked robotic systems, coordinated control of mobile autonomous agents. 1. INTRODUCTION In recent years, the topic of cooperative control and control of multi-agent systems has gained much attention. Numerous research papers (for example, Belta and Kumar (2004); Olfati-Saber and Murray (2002); Tabuada et al. (2005); Tanner et al. (2004); Ren and Beard (2004); Fax and Murray (2002, 2004))) have already been published on formation control. These papers differ in factors such as the types of agent dynamics, control strategies, formation control tasks and the information flow between the agents. This interest arises from the broad potential for applications, including formation flight, satellite clusters, advanced transportation systems, distributed sensor networks, flocking and schooling, search-and-rescue operations, competitive games, and military reconnaissance and surveillance. In this paper, agents will simply be thought of as autonomous agents including unmanned aerial vehicles, autonomous underwater vehicles, unmanned ground vehicles and micro-satellites. A formation is a group of agents moving in real 2- or 3-space, with some specified links whose distances are maintained. In this paper, we are concerned with formations in 2-space. A formation is called rigid if the distance between each pair of agents does not change over time under ideal conditions. A formation is called minimally rigid if it loses its rigidity when any one of its links is removed from the formation. In other words, a minimally rigid formation has the minimum number of links to maintain rigidity. If a formation is rigid but not minimally rigid, then it is called a redundantly rigid formation. Sensing and communication links are used for maintaining fixed distances between agents. The interconnection struc? This research was supported by TUBITAK ¨ (The Scientific and Technological Research Council of Turkey) under grant No. 108E189.
ture of sensing and communication links is called sensornetwork topology. In practice, actual agent groups cannot be expected to move exactly as a rigid formation because of sensing errors, actuation errors, actuation delays, vehicle modelling errors, etc. The ideal benchmark formation against which the performance of an actual agent formation is to be measured is called a reference formation. In reality, agents are entities with physical dimensions. For modeling purposes in this paper, agents are represented by points called point agents. Distances between all agent pairs can be held fixed by directly measuring distances between only some agents and keeping them at desired values. A distance constraint or link, is a requirement that a distance between two agents, depicted with d, be maintained through a sensing-communication link and some control strategy. Distance constraints are sometimes referred to as range or separation constraints. With enough distance constraints, the whole formation will be rigid, even without there being a distance constraint between every pair of agents. Two agents connected by a sensing-communication link are called neighbors. There are two types of control structures in formations. In the first type, the control structure is symmetric, i.e., if agent i senses and/or communicates with agent j and performs action upon the information it receives, so does agent j with agent i. A link with a symmetric control structure is represented graphically by a straight line. In the second type, the control structure is asymmetric, i.e., if agent i senses and/or communicates with agent j and performs actions upon the information it receives, then agent j does not make use of any information received from agent i although it may sense and/or communicate with agent i. A formation that has only asymmetric control structure is called a leader-follower formation. A link with an asymmetric control structure
between a leader and a follower is represented by a directed edge, or arrow, pointing from the follower to the leader, i.e., head is the leader and tail is the follower. The work in Eren et al. (2004); Olfati-Saber and Murray (2002); Eren et al. (2002) suggested an approach based on rigidity for maintaining formations of autonomous agents with sensor-network topologies that use distance information between agents, where the control structure is symmetric. Rigidity of formations with symmetric control structure that use distance information is well understood in 2-space, and there are partial results in 3-space. The appearance of the work in Eren et al. (2005) and Hendrickx et al. (2005) emphasized controlling a group of mobile autonomous agents in a formation with acyclic control structure. The graph theoretical notion, called persistence, which generalizes the notion of rigidity to directed graphs, was introduced in Hendrickx et al. (2007); C. Yu et al. (2007); Fidan et al. (2007); B.D.O. Anderson et al. (2006) to analyze the behavior of autonomous agent formations governed by unilateral distance constraints. Persistence is examined in Hendrickx et al. (2007) for two dimensions, and in J.M. Hendrickx et al. (2005); C. Yu et al. (2007, 2005) for three dimensions. A natural approach to represent the interconnection topology of a formation is a graph. Each agent is modeled as a vertex on the graph, and an edge joins vertex i to vertex j if agent j is sensing agent i and agent i is sensing agent j. A key challenge for formations of this type is stability analysis since cycles in the graph bring in a global component to each agent’s dynamics which depends on both the composition of the graph and the agent dynamics (Fax and Murray (2004)). The Laplacian of a graph, a matrix representation of the graph whose spectral properties are related to structural properties of the graph, are used in the stability analysis of interconnected agents. A control-theoretic approach to stability analysis of interconnected agents was given in Fax and Murray (2002, 2004). Establishing measures of near-periodicity would be useful in quantifying formation stability margins. In this paper, we analyze constructions to create non-3-periodic persistent formations, i.e., we study the constructions to create 3-cycle-free persistent formations. Our work on developing a general theory of creating k-cycle free rigid and persistent formations with symmetric control structures and asymmetric control structures is in progress at the time of this submission. In this paper, we only consider formations with asymmetric control structures. Thus we will use directed graphs throughout the paper. Our results on formations with symmetric control structures and undirected graphs were reported in Eren (2009). Paper is organized as follows. §II and §III give backgrounds on rigid formations and persistent formations, respectively. §IV summarizes the use of Laplacian matrix in stability analysis. Creating 3-cycle-free rigid graphs are studied in §V. The paper ends with concluding remarks in §VI. 2. RIGID FORMATIONS In this section, we briefly review rigidity and rigid formations. The notion of rigidity is useful for creating formations with symmetric control structures. On the other
hand, the notion of persistence, explained in §3, is useful for creating formations with asymmetric control structures. Persistence is based on rigidity, therefore we first start with explaining rigidity and rigid formations. A formation is rigid if the distance between each pair of agents does not change over time under ideal conditions. It is not necessary to have sensing and communication links between each pair of agents to maintain a rigid formation (Eren et al. (2002)). Distances between all agent pairs can be held fixed by directly measuring distances between only some agents and keeping them at desired values. Central to the development of rigid formations is rigid frameworks studied in mathematics and engineering for more than a century under different names such as frameworks, linkages, and mechanisms. One way of visualizing rigidity is to imagine a collection of rigid bars connected to one another by idealized ball joints, which is called a bar-joint framework. By an idealized ball joint we mean a connection between a collection of bars which imposes only the restriction that the bars share common endpoints. Now, can the bars and joints be moved in a continuous manner without changing the lengths of any of the bars, where translations and rotations do not count? If so, the framework is non-rigid; if not, it is rigid. The answer depends on factors such as which bars are connected to each other at which ball joints, bar lengths, and the dimensionality of the space in which the framework is placed. Actual physical bar-joint frameworks can be used in modeling a wide variety of physical structures, including rigid ones such as bridges as well as non-rigid structures such as organic molecules. Appropriate bar-joint frameworks representing such structures could be constructed to test the model for rigidity. However, such a concrete framework is feasible only for 2- and 3-dimensional space, and such concrete models become cumbersome as the number of bars and joints increases. The aim of rigidity theory is to develop methods for predicting rigidity without building a model. The idea of a point formation is essentially the same as the concept of a “framework” studied in mathematics as well as within the theory of structures in mechanical and civil engineering. For our purposes, a point formation Fp = (p1 , p2 , ..., pn , E) provides a natural high-level model for a set of n agents moving in real 2-space. In this context, the points pi represent the positions of agents in IR2 and the links in E label those specific agent pairs whose interagent distances are to be maintained over time. In practice actual agent positions cannot be expected to move exactly in formation because of sensing errors, vehicle modelling errors, etc. The ideal benchmark formation against which the performance of an actual agent formation is to be measured is called a reference formation. Each point formation Fp uniquely determines a graph G = (V, E) with vertex set V = {1, 2, ..., n} and edge set E, as well as a distance function δ : E → R whose value at (i, j) ∈ E is the distance between pi and pj . We say that two point formations Fp and Fq are congruent if they have the same graph and if p and q are congruent. By a trajectory of Fp , we mean a continuously parameterized, one-parameter family of points q(t) : t ≥ 0 in IR2n , which contains p. A point formation Fp is said to be rigid if the
distance between every pair of its points remains constant along any trajectory on which the lengths of all of its maintenance links in E are kept fixed. In other words, a point formation Fp is said to be rigid if rigid motion is the only kind of motion it can undergo along any trajectory on which the lengths of all links in E remain constant. Thus, if Fp is rigid, it is possible to “keep formation” by making sure that the lengths of the formation’s maintained links do not change as the formation moves. A formation is called minimally rigid if it loses its rigidity when any one of its links is removed from the formation. 2.1 Generic Rigidity “Generic rigidity” is the type of rigidity that is most useful for our purposes. In practice, actual agent groups cannot be expected to move exactly in rigid formation because of sensing, modeling, and actuation errors. With generic rigidity, the topology will be robust for maintaining formations under small perturbations. A point formation Fp is generically rigid if it is rigid for almost all choices of p in IR2n . Generic rigidity is a property of only the set of maintenance links, or the underlying graph. It does not even claim that Fp itself is rigid but only that almost all nearby points q give rigid formations Fq . The concept of generic rigidity does not depend on the precise distances between the points of Fp but examines how well the rigidity of formations can be judged by knowing the vertices and their incidences, in other words, by knowing the underlying graph. For 2-space, there is a complete combinatorial characterization of generically rigid graphs. In the theorem below, |.| is used to denote the cardinal number of a set, i.e., the number of elements in a set. Theorem 1. (Laman’s Theorem, see Whiteley (1996)). A graph G = (V, E) (where E 6= ∅, n > 1) is generically rigid in 2-dimensional space if and only if there is a subset E 0 ⊆ E satisfying the following two conditions: (1) |E 0 | = 2|V | − 3, (2) For all E 00 ⊆ E 0 , E 00 6= ∅, |E 00 | ≤ 2|V (E 00 )| − 3, where |V (E 00 )| is the number of vertices that are end-vertices of the edges in E 00 . 2.2 Henneberg Operations In this section, we present sequential techniques to create minimally rigid point formations. As noted earlier, Laman’s Theorem characterizes rigidity in 2-space. There are Henneberg operations for generating rigid classes of graphs in 2-space based on what are known as the vertex addition, edge splitting operations. First, we introduce these two operations. Before explaining these operations, we introduce some additional terminology. If (i, j) is an edge, then we say that i and j are adjacent or that j is a neighbor of i and i is a neighbor of j. The vertices i and j are incident with the edge (i, j). Two edges are adjacent if they have exactly one common end-vertex. The degree or valency of a vertex i is the number of neighbors of i. If a vertex has k neighbors, it is called a vertex of degree k or a k-valent vertex. We refer to Bollob´as (1998) and Godsil and Royle (2001) for a detailed treatment of graph theoretic terms. The first operation is the vertex addition: given a minimally rigid graph G = (V, E), we add a new
vertex i with 2 edges between i and 2 other vertices in V . The other is the edge splitting: given a minimally rigid graph G = (V, E), we remove an edge (j, k) in E and then we add a new vertex i with 3 edges by inserting two edges (i, j), (i, k) and one edge between i and one vertex (other than j, k) in V . Now we are ready to present the following theorems: Theorem 2. (Vertex addition, see Tay and Whiteley (1985)) Let G = (V, E) be a graph with a vertex i of degree 2 in 2-space; let G∗ = (V ∗ , E ∗ ) denote the subgraph obtained by deleting i and the edges incident with it. Then G is generically minimally rigid if and only if G∗ is generically minimally rigid. Theorem 3. (Edge splitting, see Tay and Whiteley (1985)) Let G = (V, E) be a graph with a vertex i of degree 3; let Vi be the set of vertices incident to i; and let G∗ = (V ∗ , E ∗ ) be the subgraph obtained by deleting i and its 3 incident edges. Then G is generically minimally rigid if and only if there is a pair j, k of vertices of Vi such that the edge (j, k) is not in E∗ and the graph G0 = (V ∗ , E ∗ ∪ (j, k)) is generically minimally rigid. 3. PERSISTENT FORMATIONS Now, we consider extensions of the notions and techniques in §2 to formations with asymmetric control structure. The task we address is maintenance of the shape of a leaderfollower formation while the formation moves. First, we give some definitions from graph theory, which are relevant to all directed point formations with leaderfollower architecture. A graph in which each edge is replaced by a directed edge is called a digraph, also called a directed graph. When there is a danger of confusion, we will call a graph, which is not a digraph, an undirected graph. A directed edge is written with an ordered pair of endvertices (i, j) representing an edge directed from i to j and drawn with an arrow from i to j, that is from the follower to the leader of that edge. Symmetric pairs of directed edges are called bidirected edges. In the context of formations, a bidirected edge is mathematically equivalent to an undirected edge in the underlying graph of a formation. In formations that have a leader-follower architecture we will only use digraphs with no bidirected edges and the results in this paper hold only for graphs with no bidirected edges. The number of edges directed into a given vertex i in a digraph G is called the in-degree of the vertex and is denoted by d− G (i). The number of edges directed out from a given vertex i in a digraph G is called the out-degree of the vertex and is denoted by d+ G (i). The out-neighborhood + NG (i) of a vertex i is {j ∈ V : (i, j) ∈ E}, and the in− neighborhood NG (i) of a vertex i is {j ∈ V : (j, i) ∈ E}. The union of out-neighborhood and in-neighborhood is the set of neighbors of i, i.e., the (open) neighborhood of i, NG (i). Any digraph in 2-space which has no more than two outgoing edges (constraint links) from any vertex is constraint consistent. We call a formation with asymmetric control structure persistent if it is rigid and constraint consistent (Hendrickx et al. (2007)). Persistence is an amalgam of two conditions, rigidity and a notion termed constraint
à with no repeated vertices aside from the start/end vertex G∗ (V ∗ , E ∗# ) is a simple cycle. In graph theory, most often “simple” q :k » »» » is implied, i.e., “cycle” means “simple cycle” and “path” » »X i qX XXX means “simple path.” The length of a cycle is the number zqj X of edges in the cycle. An N-cycle on graph G is a cycle of length N . A graph with the property that the set of all "! " ! cycle lengths has a common divisor is said to be k-periodic. (a) (b) The Laplacian matrix is a matrix representation of a graph. Together with Kirchhoff’s theorem it can be used Fig. 1. Directed vertex addition operation. to calculate the number of spanning trees for a given ∗ ∗ ∗# à graph. The Laplacian matrix can be used to find many à #G(V, G (V , E ) E) other properties of the graph. The adjacency matrix of a :q » kq »»» k 6 » simple graph is a matrix with rows and columns labeled q » iX y XXX by graph vertices, with a 1 or 0 in position (i,j) according Q jq Q XXqj to whether i and j are adjacent or not. For a simple graph Q Q with no self-loops, the adjacency matrix must have 0’s Q q sw Q "! " ! on the diagonal. For an undirected graph, the adjacency matrix is symmetric. For a digraph, it is asymmetric. The (a) (b) degree (or valency) of a vertex of a digraph is the number Fig. 2. Directed edge splitting operation. of edges leaving that vertex. The degree of a vertex v is consistence (B.D.O. Anderson et al. (2007)). Similarly, a denoted deg(v). Given a graph G with n vertices (without minimally persistent formation is that which is minimally loops or multiple edges), its Laplacian matrix L is given by L = D − A where D is the matrix with the degrees of rigid and constraint consistent. vertices along the diagonal, A is the standard adjacency Natural extensions of Henneberg operations to directed matrix. That is, it is the difference of the degree matrix and graphs exist (see Eren et al. (2005); Hendrickx et al. the adjacency matrix of the digraph. The diagonal entry (2005)). The following three operations are used in adding dii of the Laplacian matrix is the out-degree of vertex i a new agent to an existing persistent formation so that and the negative, non-diagonal entries in row i correspond the resulting formation is also persistent. Let j, k be two to the out-neighborhood of vertex i. Therefore the row distinct vertices of a minimally persistent graph G = sums of a Laplacian matrix are always zero and hence zero (V, E). A directed vertex addition consists in adding a is always an eigenvalue. The eigenvalues of a Laplacian vertex i and two directed edges (i, j) and (i, k). A reverse matrix for graphs have special properties (see for example, directed vertex addition consists in removing a vertex with Chung (1997)), some of which are that all eigenvalues are an out-degree 2 and an in-degree 0 from a minimally non-negative; the smallest eigenvalue of the Laplacian is persistent graph. Let now (j, k) be a directed edge in a zero; and its multiplicity equals the number of connected minimally persistent graph and w a distinct vertex. A components of the graph. The second eigenvalue is directly directed edge splitting consists in adding a vertex i, an related to the connectivity of the graph. edge (i, w), and replacing the edge (j, k) by (j, i) and (i, k). Let now i be a vertex with out-degree 2 and in- All eigenvalues of L are located in a disk of radius 1 in degree 1, call j the vertex left by an edge arriving at i, and the complex plane centered at 1 + j0. This can be shown k, w the other neighbors of i. The reverse directed edge by applying Gershgorin’s theorem to the rows of L (Fax splitting operation consists in removing i and its incident and Murray (2004)). If G is aperiodic, then no eigenvalues edges, and adding either (j, k) or (j, w) (k and w being (other than the zero eigenvalue) will lie on the boundary interchangeable) in such a way that the graph obtained is of the Gershgorin disk. If G is k-periodic, then L has k on the boundary with an angular spacing of minimally rigid. The directed vertex addition and edge eigenvalues 2π . If G is an undirected graph, all eigenvalues of L are k splitting operations preserve minimal persistence, and real. so do the reverse directed vertex addition and reverse directed edge splitting operations. These two operations Example 4.1. Let us consider the graph in Fig. 3. Its are depicted on formation graphs in Figs. 1 and 2. The Laplacian matrix is given as: third operation which will complete the constructions is edge reversal: edge (a, b) is reversed to (b, a), if for the 2 0 −1 −1 + in-vertex b, NG (b) < 2. −1 1 0 0 0 0 1 −1 4. GRAPH LAPLACIAN 0 −1 0 1 à #G(V, E)
A path in a directed graph is a directed sequence of vertices, based on the direction of edges, such that from each of its vertices there is an edge to the next vertex in the sequence. The first vertex is called the start vertex and the last vertex is called the end vertex. A cycle is a path such that the start vertex and end vertex are the same. The choice of the start vertex in a cycle is arbitrary. A path with no repeated vertices is called a simple path, and cycle
Formation structure affects eigenvalue placement, thus the Laplacian eigenvalues are used to analyze formation stability on local level (Fax and Murray (2004)). For example, the complete graph is one where every possible edge exists. In this case, the eigenvalues of a graph with N vertices are zero and 1 + (1/N − 1), where the latter repeated N − 1 times. Undirected graph will have an eigenvalue at 2, due to its periodicity, and all other eigenvalues will be real.
2q J ] J 3q J © *HH J ©© H 1© À H jJq 4 q Fig. 3. Sample graph to demonstrate the Laplacian matrix. An acyclic graph has all eigenvalues at λ = 1. A single directed cycle has eigenvalues at 1 − ej(i−1)/2π , i ∈ [1, N ]. Eigenvalue placement is related to the rate of mixing of information through the formation (Fax and Murray (2004)). When the graph is highly connected, the global component of an individual agent’s dynamics is quickly averaged out through the rest of the graph, and so has only a minor effect on stability. When the graph is periodic, the global component of the dynamics introduces periodic forcing of the agent, and the rest of the network never averages it out. Thus aperiodicity has emerged as a desirable property of formation interconnection topologies (Fax and Murray (2004)). 5. RESULTS: 3-CYCLE-FREE PERSISTENT FORMATIONS IN 2-SPACE Aperiodicity is a desirable property of formation interconnection topologies as stated in the last section. Now we will explore creating non-3-periodic minimally persistent digraphs. To do that, we consider creating 3-cycle-free persistent digraphs. Such persistent digraphs do not have any 3-cycles. The distance between vertices v and w, denoted by d(v, w), is the length of the shortest path directed from v to w. We have the following theorem: Theorem 4. (3-cycle-free directed vertex addition) Let a digraph G = (V, E) be given. A vertex i of degree 2 is inserted into G by adding the edges (i, j) and (i, k). Let G∗ = (V ∗ , E ∗ ) denote the resultant digraph, i.e., V ∗ = V ∪ {i} and E ∗ = E ∪ {(i, j), (i, k)}. G = (V, E) is 3-cycle-free minimally persistent if and only if G∗ = (V ∗ , E ∗ ) is 3cycle-free minimally persistent. Proof. Firstly, assume that G = (V, E) is 3-cycle-free minimally persistent. All newly added paths include vertex i and start from vertex i, since both (i, j) and (i, k) are directed away from vertex i. Any new path starting from vertex i will not come back to vertex i again, since indegree of vertex i is zero. Therefore the resultant digraph G∗ = (V ∗ , E ∗ ) does not include a 3-cycle. Furthermore G∗ is minimally rigid since the operation is vertex addition by Theorem 2, and it is constraint consistent. Thus G∗ = (V ∗ , E ∗ ) is 3-cycle-free minimally persistent. Now assume that G∗ = (V ∗ , E ∗ ) is 3-cycle-free minimally persistent. When we remove the vertex i and its incident edges (i, j) and (i, k), the resultant digraph is still minimally persistent (from vertex addition in reverse direction by Theorem 2 and constraint consistency still hold). Furthermore, deleting edges does not create a new cycle of any length. Thus the resultant digraph G = (V, E) is 3-cycle-free minimally persistent. 2
iq Q J ]Q J QQ vq J Q X y XXJX Q Y H ©©H Q H Jq XXX j q© À © Hk ¼ Q sq X * © » HH © » p » © H q» H j© »»» » 9 w Fig. 4. Counterexample for directed edge splitting operation in reverse direction.
vq X y XX Y ©H H XXX HH j q© ©© q k XX q ¼ * © » p HH © »» © H q» H j© »»» » 9 w Fig. 5. When i is removed from the digraph shown in Fig. 4, there are two possibilities to insert the new edge: namely (k, j) and (k, p). These two result in 3-cycles, (k, j, w) and (k, p, w). If edge reversal is applied on (k, p) resulting into (p, k), then the cycle (k, p, w) is avoided. Now we consider edge splitting operation for 3-cycle-free minimally rigid digraphs. First we consider removing a vertex of degree 3. We have the following proposition: Proposition 5. (3-cycle-free directed edge splitting in reverse direction) Let G∗ = (V ∗ , E ∗ ) be a 3-cycle-free minimally persistent digraph, where V ∗ = V ∪ {i} and E ∗ = E ∪ {(i, j), (k, i), (i, p)}. Assume that we remove vertex i and its incident edges (i, j), (k, i), (i, p). Then ˆ is 3-cycle-free minithere is no guarantee that G(V, E) ˆ = E ∪ {any one of mally persistent digraph, where E the pairs (k, j), (k, p)}, unless edge reversal operation is applied after edge splitting. ˆ is 3-cycle-free minProof. Let us assume that G(V, E) ˆ = E ∪ {any one of imally persistent digraph, where E the pairs (k, j), (k, p)}. Now let us prove that this is not possible by giving a counterexample. Consider the digraph G∗ (V ∗ , E ∗ ) shown in Fig. 4. It is a 3-cycle-free minimally persistent digraph. Assume that we remove vertex i with its incident edges (i, j), (k, i), (i, p) and we obtain G(V, E) as shown in Fig. 5. Then, by applying reverse of directed edge splitting operation, we insert the edges (k, j), or (k, p) ˆ1 ), G2 (V, E ˆ2 ) separately into G(V, E) to obtain G1 (V, E ˆ ˆ respectively. G1 (V, E1 ) has a 3-cycle (j, w, k). G2 (V, E2 ) has a 3-cycle (w, k, p). One of these two cycles can be avoided if (k, p) is reversed to (p, k) by applying edge reversal operation. 2 The work on additional conditions to achieve directed edge splitting in reverse direction without a need for edge reversal operation is currently in progress. Now we consider edge splitting by inserting a vertex of degree 3 into a 3-cycle-free minimally persistent digraph. By using Theorem 3, the analysis may be completed to establish 3-cycle-free minimal persistence in edge splitting.
At the time of submission this analysis was incomplete. However, the following conjecture has been derived. Conjecture 6. (3-cycle-free directed edge splitting) Let G = (V, E) be a 3-cycle-free minimally persistent digraph. Assume that (j, k) ∈ E . Suppose that there exists a vertex w ∈ V where d(w, j) ≥ 2, such that a vertex i of degree 3 is added to G by removing (j, k) and inserting (j, i), (i, k), and a third edge (i, w) between i and w. Let the resultant digraph be denoted by G∗ (V ∗ , E ∗ ) where V ∗ = V ∪ {i} and E ∗ = E ∪ {(j, i), (i, k), (i, w)}. Then G∗ is 3-cycle-free minimally persistent digraph. REFERENCES B.D.O. Anderson, C. Yu, and B. Fidan (2007). Information architecture and control design for rigid formations. In Proceedings of the 26th Chinese Control Conference, 2–10. China. B.D.O. Anderson, C. Yu, B. Fidan, and J. M. Hendrickx (2006). Control and information architectures for formations. In Proceedings of the IEEE Conference on Control Applications, 1127–1138. Belta, C. and Kumar, V. (2004). Abstraction and control for groups of robots. IEEE Transactions on Robotics, 20(5), 865–875. Bollob´as, B. (1998). Modern Graph Theory. Springer Verlag, New York, New York. C. Yu, J. M. Hendrickx, B. Fidan, B.D.O. Anderson, and V.D. Blondel (2007). Three and higher dimensional autonomous formations: Rigidity, persistence and structural persistence. Automatica, 43(3), 387–402. C. Yu, J.M. Hendrickx, B. Fidan, and B.D.O. Anderson (2005). Structural persistence of three-dimensional autonomous formations. In Proceedings of the 1st International Workshop on Multi-Agent Robotic Systems. Barcelona, Spain. Chung, F. (1997). Spectral Graph Theory. American Mathematical Society. Eren, T., B.D.O. Anderson, A.S. Morse, Whiteley, W., and P.N. Belhumeur (2004). Operations on rigid formations of autonomous agents. Communications in Information and Systems, 3(4). Eren, T., P.N. Belhumeur, B.D.O. Anderson, and A.S. Morse (2002). A framework for maintaining formations based on rigidity. In Proceedings of the 15th IFAC World Congress, Barcelona, Spain, 2752–2757. Eren, T. (2009). 3-cycle-free rigidity for multi-agent control systems. Submitted to the 17th Mediterranean Conference on Control and Automation. Eren, T., Whiteley, W., B. D. O. Anderson, Morse, A.S., and Belhumeur, P.N. (2005). Information structures to secure control of rigid formations with leader-follower architecture. In Proceedings of the American Control Conference, 2966–2971. Portland, Oregon. Fax, J.A. and Murray, R.M. (2002). Graph laplacians and stabilization of vehicle formations. In Proceedings of the 15th IFAC World Congress. Barcelona, Spain. Fax, J.A. and Murray, R.M. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49(9), 1465–1476. Fidan, B., C.Yu, and B. D. O. Anderson (2007). Acquiring and maintaining persistence of autonomous multivehicle formations. IET Control Theory and Applications, 1(2), 452–460.
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ture of sensing and communication links is called sensornetwork topology. In practice, actual agent groups cannot be expected to move exactly as a rigid formation because of sensing errors, actuation errors, actuation delays, vehicle modelling errors, etc. The ideal benchmark formation against which the performance of an actual agent formation is to be measured is called a reference formation. In reality, agents are entities with physical dimensions. For modeling purposes in this paper, agents are represented by points called point agents. Distances between all agent pairs can be held fixed by directly measuring distances between only some agents and keeping them at desired values. A distance constraint or link, is a requirement that a distance between two agents, depicted with d, be maintained through a sensing-communication link and some control strategy. Distance constraints are sometimes referred to as range or separation constraints. With enough distance constraints, the whole formation will be rigid, even without there being a distance constraint between every pair of agents. Two agents connected by a sensing-communication link are called neighbors. There are two types of control structures in formations. In the first type, the control structure is symmetric, i.e., if agent i senses and/or communicates with agent j and performs action upon the information it receives, so does agent j with agent i. A link with a symmetric control structure is represented graphically by a straight line. In the second type, the control structure is asymmetric, i.e., if agent i senses and/or communicates with agent j and performs actions upon the information it receives, then agent j does not make use of any information received from agent i although it may sense and/or communicate with agent i. A formation that has only asymmetric control structure is called a leader-follower formation. A link with an asymmetric control structure
between a leader and a follower is represented by a directed edge, or arrow, pointing from the follower to the leader, i.e., head is the leader and tail is the follower. The work in Eren et al. (2004); Olfati-Saber and Murray (2002); Eren et al. (2002) suggested an approach based on rigidity for maintaining formations of autonomous agents with sensor-network topologies that use distance information between agents, where the control structure is symmetric. Rigidity of formations with symmetric control structure that use distance information is well understood in 2-space, and there are partial results in 3-space. The appearance of the work in Eren et al. (2005) and Hendrickx et al. (2005) emphasized controlling a group of mobile autonomous agents in a formation with acyclic control structure. The graph theoretical notion, called persistence, which generalizes the notion of rigidity to directed graphs, was introduced in Hendrickx et al. (2007); C. Yu et al. (2007); Fidan et al. (2007); B.D.O. Anderson et al. (2006) to analyze the behavior of autonomous agent formations governed by unilateral distance constraints. Persistence is examined in Hendrickx et al. (2007) for two dimensions, and in J.M. Hendrickx et al. (2005); C. Yu et al. (2007, 2005) for three dimensions. A natural approach to represent the interconnection topology of a formation is a graph. Each agent is modeled as a vertex on the graph, and an edge joins vertex i to vertex j if agent j is sensing agent i and agent i is sensing agent j. A key challenge for formations of this type is stability analysis since cycles in the graph bring in a global component to each agent’s dynamics which depends on both the composition of the graph and the agent dynamics (Fax and Murray (2004)). The Laplacian of a graph, a matrix representation of the graph whose spectral properties are related to structural properties of the graph, are used in the stability analysis of interconnected agents. A control-theoretic approach to stability analysis of interconnected agents was given in Fax and Murray (2002, 2004). Establishing measures of near-periodicity would be useful in quantifying formation stability margins. In this paper, we analyze constructions to create non-3-periodic persistent formations, i.e., we study the constructions to create 3-cycle-free persistent formations. Our work on developing a general theory of creating k-cycle free rigid and persistent formations with symmetric control structures and asymmetric control structures is in progress at the time of this submission. In this paper, we only consider formations with asymmetric control structures. Thus we will use directed graphs throughout the paper. Our results on formations with symmetric control structures and undirected graphs were reported in Eren (2009). Paper is organized as follows. §II and §III give backgrounds on rigid formations and persistent formations, respectively. §IV summarizes the use of Laplacian matrix in stability analysis. Creating 3-cycle-free rigid graphs are studied in §V. The paper ends with concluding remarks in §VI. 2. RIGID FORMATIONS In this section, we briefly review rigidity and rigid formations. The notion of rigidity is useful for creating formations with symmetric control structures. On the other
hand, the notion of persistence, explained in §3, is useful for creating formations with asymmetric control structures. Persistence is based on rigidity, therefore we first start with explaining rigidity and rigid formations. A formation is rigid if the distance between each pair of agents does not change over time under ideal conditions. It is not necessary to have sensing and communication links between each pair of agents to maintain a rigid formation (Eren et al. (2002)). Distances between all agent pairs can be held fixed by directly measuring distances between only some agents and keeping them at desired values. Central to the development of rigid formations is rigid frameworks studied in mathematics and engineering for more than a century under different names such as frameworks, linkages, and mechanisms. One way of visualizing rigidity is to imagine a collection of rigid bars connected to one another by idealized ball joints, which is called a bar-joint framework. By an idealized ball joint we mean a connection between a collection of bars which imposes only the restriction that the bars share common endpoints. Now, can the bars and joints be moved in a continuous manner without changing the lengths of any of the bars, where translations and rotations do not count? If so, the framework is non-rigid; if not, it is rigid. The answer depends on factors such as which bars are connected to each other at which ball joints, bar lengths, and the dimensionality of the space in which the framework is placed. Actual physical bar-joint frameworks can be used in modeling a wide variety of physical structures, including rigid ones such as bridges as well as non-rigid structures such as organic molecules. Appropriate bar-joint frameworks representing such structures could be constructed to test the model for rigidity. However, such a concrete framework is feasible only for 2- and 3-dimensional space, and such concrete models become cumbersome as the number of bars and joints increases. The aim of rigidity theory is to develop methods for predicting rigidity without building a model. The idea of a point formation is essentially the same as the concept of a “framework” studied in mathematics as well as within the theory of structures in mechanical and civil engineering. For our purposes, a point formation Fp = (p1 , p2 , ..., pn , E) provides a natural high-level model for a set of n agents moving in real 2-space. In this context, the points pi represent the positions of agents in IR2 and the links in E label those specific agent pairs whose interagent distances are to be maintained over time. In practice actual agent positions cannot be expected to move exactly in formation because of sensing errors, vehicle modelling errors, etc. The ideal benchmark formation against which the performance of an actual agent formation is to be measured is called a reference formation. Each point formation Fp uniquely determines a graph G = (V, E) with vertex set V = {1, 2, ..., n} and edge set E, as well as a distance function δ : E → R whose value at (i, j) ∈ E is the distance between pi and pj . We say that two point formations Fp and Fq are congruent if they have the same graph and if p and q are congruent. By a trajectory of Fp , we mean a continuously parameterized, one-parameter family of points q(t) : t ≥ 0 in IR2n , which contains p. A point formation Fp is said to be rigid if the
distance between every pair of its points remains constant along any trajectory on which the lengths of all of its maintenance links in E are kept fixed. In other words, a point formation Fp is said to be rigid if rigid motion is the only kind of motion it can undergo along any trajectory on which the lengths of all links in E remain constant. Thus, if Fp is rigid, it is possible to “keep formation” by making sure that the lengths of the formation’s maintained links do not change as the formation moves. A formation is called minimally rigid if it loses its rigidity when any one of its links is removed from the formation. 2.1 Generic Rigidity “Generic rigidity” is the type of rigidity that is most useful for our purposes. In practice, actual agent groups cannot be expected to move exactly in rigid formation because of sensing, modeling, and actuation errors. With generic rigidity, the topology will be robust for maintaining formations under small perturbations. A point formation Fp is generically rigid if it is rigid for almost all choices of p in IR2n . Generic rigidity is a property of only the set of maintenance links, or the underlying graph. It does not even claim that Fp itself is rigid but only that almost all nearby points q give rigid formations Fq . The concept of generic rigidity does not depend on the precise distances between the points of Fp but examines how well the rigidity of formations can be judged by knowing the vertices and their incidences, in other words, by knowing the underlying graph. For 2-space, there is a complete combinatorial characterization of generically rigid graphs. In the theorem below, |.| is used to denote the cardinal number of a set, i.e., the number of elements in a set. Theorem 1. (Laman’s Theorem, see Whiteley (1996)). A graph G = (V, E) (where E 6= ∅, n > 1) is generically rigid in 2-dimensional space if and only if there is a subset E 0 ⊆ E satisfying the following two conditions: (1) |E 0 | = 2|V | − 3, (2) For all E 00 ⊆ E 0 , E 00 6= ∅, |E 00 | ≤ 2|V (E 00 )| − 3, where |V (E 00 )| is the number of vertices that are end-vertices of the edges in E 00 . 2.2 Henneberg Operations In this section, we present sequential techniques to create minimally rigid point formations. As noted earlier, Laman’s Theorem characterizes rigidity in 2-space. There are Henneberg operations for generating rigid classes of graphs in 2-space based on what are known as the vertex addition, edge splitting operations. First, we introduce these two operations. Before explaining these operations, we introduce some additional terminology. If (i, j) is an edge, then we say that i and j are adjacent or that j is a neighbor of i and i is a neighbor of j. The vertices i and j are incident with the edge (i, j). Two edges are adjacent if they have exactly one common end-vertex. The degree or valency of a vertex i is the number of neighbors of i. If a vertex has k neighbors, it is called a vertex of degree k or a k-valent vertex. We refer to Bollob´as (1998) and Godsil and Royle (2001) for a detailed treatment of graph theoretic terms. The first operation is the vertex addition: given a minimally rigid graph G = (V, E), we add a new
vertex i with 2 edges between i and 2 other vertices in V . The other is the edge splitting: given a minimally rigid graph G = (V, E), we remove an edge (j, k) in E and then we add a new vertex i with 3 edges by inserting two edges (i, j), (i, k) and one edge between i and one vertex (other than j, k) in V . Now we are ready to present the following theorems: Theorem 2. (Vertex addition, see Tay and Whiteley (1985)) Let G = (V, E) be a graph with a vertex i of degree 2 in 2-space; let G∗ = (V ∗ , E ∗ ) denote the subgraph obtained by deleting i and the edges incident with it. Then G is generically minimally rigid if and only if G∗ is generically minimally rigid. Theorem 3. (Edge splitting, see Tay and Whiteley (1985)) Let G = (V, E) be a graph with a vertex i of degree 3; let Vi be the set of vertices incident to i; and let G∗ = (V ∗ , E ∗ ) be the subgraph obtained by deleting i and its 3 incident edges. Then G is generically minimally rigid if and only if there is a pair j, k of vertices of Vi such that the edge (j, k) is not in E∗ and the graph G0 = (V ∗ , E ∗ ∪ (j, k)) is generically minimally rigid. 3. PERSISTENT FORMATIONS Now, we consider extensions of the notions and techniques in §2 to formations with asymmetric control structure. The task we address is maintenance of the shape of a leaderfollower formation while the formation moves. First, we give some definitions from graph theory, which are relevant to all directed point formations with leaderfollower architecture. A graph in which each edge is replaced by a directed edge is called a digraph, also called a directed graph. When there is a danger of confusion, we will call a graph, which is not a digraph, an undirected graph. A directed edge is written with an ordered pair of endvertices (i, j) representing an edge directed from i to j and drawn with an arrow from i to j, that is from the follower to the leader of that edge. Symmetric pairs of directed edges are called bidirected edges. In the context of formations, a bidirected edge is mathematically equivalent to an undirected edge in the underlying graph of a formation. In formations that have a leader-follower architecture we will only use digraphs with no bidirected edges and the results in this paper hold only for graphs with no bidirected edges. The number of edges directed into a given vertex i in a digraph G is called the in-degree of the vertex and is denoted by d− G (i). The number of edges directed out from a given vertex i in a digraph G is called the out-degree of the vertex and is denoted by d+ G (i). The out-neighborhood + NG (i) of a vertex i is {j ∈ V : (i, j) ∈ E}, and the in− neighborhood NG (i) of a vertex i is {j ∈ V : (j, i) ∈ E}. The union of out-neighborhood and in-neighborhood is the set of neighbors of i, i.e., the (open) neighborhood of i, NG (i). Any digraph in 2-space which has no more than two outgoing edges (constraint links) from any vertex is constraint consistent. We call a formation with asymmetric control structure persistent if it is rigid and constraint consistent (Hendrickx et al. (2007)). Persistence is an amalgam of two conditions, rigidity and a notion termed constraint
à with no repeated vertices aside from the start/end vertex G∗ (V ∗ , E ∗# ) is a simple cycle. In graph theory, most often “simple” q :k » »» » is implied, i.e., “cycle” means “simple cycle” and “path” » »X i qX XXX means “simple path.” The length of a cycle is the number zqj X of edges in the cycle. An N-cycle on graph G is a cycle of length N . A graph with the property that the set of all "! " ! cycle lengths has a common divisor is said to be k-periodic. (a) (b) The Laplacian matrix is a matrix representation of a graph. Together with Kirchhoff’s theorem it can be used Fig. 1. Directed vertex addition operation. to calculate the number of spanning trees for a given ∗ ∗ ∗# à graph. The Laplacian matrix can be used to find many à #G(V, G (V , E ) E) other properties of the graph. The adjacency matrix of a :q » kq »»» k 6 » simple graph is a matrix with rows and columns labeled q » iX y XXX by graph vertices, with a 1 or 0 in position (i,j) according Q jq Q XXqj to whether i and j are adjacent or not. For a simple graph Q Q with no self-loops, the adjacency matrix must have 0’s Q q sw Q "! " ! on the diagonal. For an undirected graph, the adjacency matrix is symmetric. For a digraph, it is asymmetric. The (a) (b) degree (or valency) of a vertex of a digraph is the number Fig. 2. Directed edge splitting operation. of edges leaving that vertex. The degree of a vertex v is consistence (B.D.O. Anderson et al. (2007)). Similarly, a denoted deg(v). Given a graph G with n vertices (without minimally persistent formation is that which is minimally loops or multiple edges), its Laplacian matrix L is given by L = D − A where D is the matrix with the degrees of rigid and constraint consistent. vertices along the diagonal, A is the standard adjacency Natural extensions of Henneberg operations to directed matrix. That is, it is the difference of the degree matrix and graphs exist (see Eren et al. (2005); Hendrickx et al. the adjacency matrix of the digraph. The diagonal entry (2005)). The following three operations are used in adding dii of the Laplacian matrix is the out-degree of vertex i a new agent to an existing persistent formation so that and the negative, non-diagonal entries in row i correspond the resulting formation is also persistent. Let j, k be two to the out-neighborhood of vertex i. Therefore the row distinct vertices of a minimally persistent graph G = sums of a Laplacian matrix are always zero and hence zero (V, E). A directed vertex addition consists in adding a is always an eigenvalue. The eigenvalues of a Laplacian vertex i and two directed edges (i, j) and (i, k). A reverse matrix for graphs have special properties (see for example, directed vertex addition consists in removing a vertex with Chung (1997)), some of which are that all eigenvalues are an out-degree 2 and an in-degree 0 from a minimally non-negative; the smallest eigenvalue of the Laplacian is persistent graph. Let now (j, k) be a directed edge in a zero; and its multiplicity equals the number of connected minimally persistent graph and w a distinct vertex. A components of the graph. The second eigenvalue is directly directed edge splitting consists in adding a vertex i, an related to the connectivity of the graph. edge (i, w), and replacing the edge (j, k) by (j, i) and (i, k). Let now i be a vertex with out-degree 2 and in- All eigenvalues of L are located in a disk of radius 1 in degree 1, call j the vertex left by an edge arriving at i, and the complex plane centered at 1 + j0. This can be shown k, w the other neighbors of i. The reverse directed edge by applying Gershgorin’s theorem to the rows of L (Fax splitting operation consists in removing i and its incident and Murray (2004)). If G is aperiodic, then no eigenvalues edges, and adding either (j, k) or (j, w) (k and w being (other than the zero eigenvalue) will lie on the boundary interchangeable) in such a way that the graph obtained is of the Gershgorin disk. If G is k-periodic, then L has k on the boundary with an angular spacing of minimally rigid. The directed vertex addition and edge eigenvalues 2π . If G is an undirected graph, all eigenvalues of L are k splitting operations preserve minimal persistence, and real. so do the reverse directed vertex addition and reverse directed edge splitting operations. These two operations Example 4.1. Let us consider the graph in Fig. 3. Its are depicted on formation graphs in Figs. 1 and 2. The Laplacian matrix is given as: third operation which will complete the constructions is edge reversal: edge (a, b) is reversed to (b, a), if for the 2 0 −1 −1 + in-vertex b, NG (b) < 2. −1 1 0 0 0 0 1 −1 4. GRAPH LAPLACIAN 0 −1 0 1 à #G(V, E)
A path in a directed graph is a directed sequence of vertices, based on the direction of edges, such that from each of its vertices there is an edge to the next vertex in the sequence. The first vertex is called the start vertex and the last vertex is called the end vertex. A cycle is a path such that the start vertex and end vertex are the same. The choice of the start vertex in a cycle is arbitrary. A path with no repeated vertices is called a simple path, and cycle
Formation structure affects eigenvalue placement, thus the Laplacian eigenvalues are used to analyze formation stability on local level (Fax and Murray (2004)). For example, the complete graph is one where every possible edge exists. In this case, the eigenvalues of a graph with N vertices are zero and 1 + (1/N − 1), where the latter repeated N − 1 times. Undirected graph will have an eigenvalue at 2, due to its periodicity, and all other eigenvalues will be real.
2q J ] J 3q J © *HH J ©© H 1© À H jJq 4 q Fig. 3. Sample graph to demonstrate the Laplacian matrix. An acyclic graph has all eigenvalues at λ = 1. A single directed cycle has eigenvalues at 1 − ej(i−1)/2π , i ∈ [1, N ]. Eigenvalue placement is related to the rate of mixing of information through the formation (Fax and Murray (2004)). When the graph is highly connected, the global component of an individual agent’s dynamics is quickly averaged out through the rest of the graph, and so has only a minor effect on stability. When the graph is periodic, the global component of the dynamics introduces periodic forcing of the agent, and the rest of the network never averages it out. Thus aperiodicity has emerged as a desirable property of formation interconnection topologies (Fax and Murray (2004)). 5. RESULTS: 3-CYCLE-FREE PERSISTENT FORMATIONS IN 2-SPACE Aperiodicity is a desirable property of formation interconnection topologies as stated in the last section. Now we will explore creating non-3-periodic minimally persistent digraphs. To do that, we consider creating 3-cycle-free persistent digraphs. Such persistent digraphs do not have any 3-cycles. The distance between vertices v and w, denoted by d(v, w), is the length of the shortest path directed from v to w. We have the following theorem: Theorem 4. (3-cycle-free directed vertex addition) Let a digraph G = (V, E) be given. A vertex i of degree 2 is inserted into G by adding the edges (i, j) and (i, k). Let G∗ = (V ∗ , E ∗ ) denote the resultant digraph, i.e., V ∗ = V ∪ {i} and E ∗ = E ∪ {(i, j), (i, k)}. G = (V, E) is 3-cycle-free minimally persistent if and only if G∗ = (V ∗ , E ∗ ) is 3cycle-free minimally persistent. Proof. Firstly, assume that G = (V, E) is 3-cycle-free minimally persistent. All newly added paths include vertex i and start from vertex i, since both (i, j) and (i, k) are directed away from vertex i. Any new path starting from vertex i will not come back to vertex i again, since indegree of vertex i is zero. Therefore the resultant digraph G∗ = (V ∗ , E ∗ ) does not include a 3-cycle. Furthermore G∗ is minimally rigid since the operation is vertex addition by Theorem 2, and it is constraint consistent. Thus G∗ = (V ∗ , E ∗ ) is 3-cycle-free minimally persistent. Now assume that G∗ = (V ∗ , E ∗ ) is 3-cycle-free minimally persistent. When we remove the vertex i and its incident edges (i, j) and (i, k), the resultant digraph is still minimally persistent (from vertex addition in reverse direction by Theorem 2 and constraint consistency still hold). Furthermore, deleting edges does not create a new cycle of any length. Thus the resultant digraph G = (V, E) is 3-cycle-free minimally persistent. 2
iq Q J ]Q J QQ vq J Q X y XXJX Q Y H ©©H Q H Jq XXX j q© À © Hk ¼ Q sq X * © » HH © » p » © H q» H j© »»» » 9 w Fig. 4. Counterexample for directed edge splitting operation in reverse direction.
vq X y XX Y ©H H XXX HH j q© ©© q k XX q ¼ * © » p HH © »» © H q» H j© »»» » 9 w Fig. 5. When i is removed from the digraph shown in Fig. 4, there are two possibilities to insert the new edge: namely (k, j) and (k, p). These two result in 3-cycles, (k, j, w) and (k, p, w). If edge reversal is applied on (k, p) resulting into (p, k), then the cycle (k, p, w) is avoided. Now we consider edge splitting operation for 3-cycle-free minimally rigid digraphs. First we consider removing a vertex of degree 3. We have the following proposition: Proposition 5. (3-cycle-free directed edge splitting in reverse direction) Let G∗ = (V ∗ , E ∗ ) be a 3-cycle-free minimally persistent digraph, where V ∗ = V ∪ {i} and E ∗ = E ∪ {(i, j), (k, i), (i, p)}. Assume that we remove vertex i and its incident edges (i, j), (k, i), (i, p). Then ˆ is 3-cycle-free minithere is no guarantee that G(V, E) ˆ = E ∪ {any one of mally persistent digraph, where E the pairs (k, j), (k, p)}, unless edge reversal operation is applied after edge splitting. ˆ is 3-cycle-free minProof. Let us assume that G(V, E) ˆ = E ∪ {any one of imally persistent digraph, where E the pairs (k, j), (k, p)}. Now let us prove that this is not possible by giving a counterexample. Consider the digraph G∗ (V ∗ , E ∗ ) shown in Fig. 4. It is a 3-cycle-free minimally persistent digraph. Assume that we remove vertex i with its incident edges (i, j), (k, i), (i, p) and we obtain G(V, E) as shown in Fig. 5. Then, by applying reverse of directed edge splitting operation, we insert the edges (k, j), or (k, p) ˆ1 ), G2 (V, E ˆ2 ) separately into G(V, E) to obtain G1 (V, E ˆ ˆ respectively. G1 (V, E1 ) has a 3-cycle (j, w, k). G2 (V, E2 ) has a 3-cycle (w, k, p). One of these two cycles can be avoided if (k, p) is reversed to (p, k) by applying edge reversal operation. 2 The work on additional conditions to achieve directed edge splitting in reverse direction without a need for edge reversal operation is currently in progress. Now we consider edge splitting by inserting a vertex of degree 3 into a 3-cycle-free minimally persistent digraph. By using Theorem 3, the analysis may be completed to establish 3-cycle-free minimal persistence in edge splitting.
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