Studying Graph Connectivity

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1 Studying Graph Connectivity Freeman Yufei Huang July 1, 2002 Submitted for CISC-871 Instructor: Dr. Robin Dawes

2 Studying Graph Connectivity Freeman Yufei Huang Submitted July 1, 2002 for CISC-871 In some sense, the major task of computing science is to abstract practical problems into mathematical models, which can be solved by computing devices. In modeling, expressiveness and complexity are always two opposite factors between which designers have to make painful decisions for trade-offs. While researchers try to make models that can describe as many problems as possible, they usually end up with reaching or even exceeding the upper limit of the computing devices. Graph is such a challenging example. Although graph theory has been developing for several decades, researchers are still improving algorithms to degrade the complexity when applying graph properties to solve practical problems, and to keep applying them to new problems. 1. Graph Connectivity in General One of the well-studied graph properties is connectivity, which I am still interested in. In lectures of the course CISC-871 we covered the simplest case of vertex connectivity, namely cut vertex and 2-connected graph. The general case of graph connectivity is usually expressed into two aspects, vertex connectivity and edge connectivity (West, Chapter 4). Researchers study graph connectivity by observing the change of connectivity while removing vertices or edges. 1.1 Notation G(V, E) represents a undirected graph G with its vertex set V and edge set E, where an edge e ij E is a unordered pair (v i v j ), where v i, v j V.

3 D(V, E) represents a directed graph (digraph) D with its vertex set V and edge set E, where an edge e ij E is an ordered pair (v i v j ), where v i, v j V. S represents the cardinality of set S. K i represents a complete graph with V = i. 1.2 Vertex Connectivity A vertex cut of G(V, E) is a subset V c V, such that G-V c has more than one component. G is said to be k-connected graph if min V c = k. Menger s theorem, which is one of the cornerstones of graph theory, describes an important characteristic of k-connected graph. It can be informally explained as: G is k-connected iff there exists k disjoint paths between any two vertices of G. For the 1-connected case, min V c = 1. The only vertex in a minimal vertex cut is called cut vertex. For the 2-connected case, a graph G of more than two vertices is 2-connected iff any two vertices of G are connected by two disjoint paths, i.e. there is a cycle going through these two vertices. A 2-connected graph has no cut vertex. A block of a graph is a maximal connected sub-graph with no cut-vertex. Two blocks of a graph share at most one vertex. A block can be K 1, a cut edge (see 1.3), or a 2- connected graph. When we extend the vertex connectivity to digraph, we can get the same results if we observe the strong connectivity in digraphs instead of simple connectivity in undirected graphs. A digraph is strongly connected if there exists a path from each vertex to every other vertex. Strongly connected digraphs are similar to 2-edgeconnected graphs (see 1.3). Strong components of a digraph D are the maximal strongly connected subgraphs of D. Strong components of a digraph behave very much like blocks of a graph. Two strong components of a digraph share no vertex.

4 1.3 Edge Connectivity A disconnecting edge set of G(V, E) is a subset E d E, such that G- E d has more than one component. G is said to be k-edge-connected graph if min E d = k. Similar to vertex connectivity, a 1-edge-connected graph has some cut edges, which are the elements of minimal disconnecting edge sets. 2-edge-connected graphs have no cut edge. Similarly, we can also extend edge connectivity to digraphs. Actually for edge connectivity researchers pay more attention to digraphs and use another term for it: reachability. 2. Digraph Reachability In a digraph D(V, E), when there exists a path from v i to v j, we say v j is reachable from v i. We define a reachability relation R on V as: v i Rv j iff either i = j or v j is reachable from v i. Reachability is reflexive and transitive. Strongly connected digraph can be defined by reachability: a digraph is strongly connected if any vertex is reachable from any others. In strongly connected digraphs, reachability relation is also symmetric, making it an equivalence relation. We call it mutual reachability. 2.1 Mutual Reachability and Strong Components In an arbitrary digraph D(V, E), mutual reachability partitions V into equivalence classes V 1,, V n. Each component (V k, E k ) is a strongly connected graph, i.e. strong component, where k=1,, n, and E k E with all endpoints of its edges in V k. For the convenience in further discussion, we introduce some notation and terminology. E S = k=1,, n E k is the set of all edges falling in either strong

5 component. E D = E-E S is the set of edges that are not in any strong component. Parallel edges between strong components are edges in E D that have the same starting component and the same ending component. Condensed graph D*=(V*, E*): V* is a set of points p 1,,p n corresponding to strong components of D, E* is a set of ordered pairs of such points, p h p k E* iff at least one edge v i v j E D such that v i V h and v j V k. Informally speaking, condensed graph is constructed by condensing the strong components of a digraph into vertices, then eliminating redundant parallel edges. Complete sequence: a finite path in a digraph, which has the same initial and terminal vertex and goes through every vertex at least once. Every strong component has at least one complete sequence. There is one of minimum length. Superfluous edge: v i1 V k1, v i2 V k2, v i3 V k3, if v i1 Rv i2, v i2 Rv i3, and v i1 v i3 E, then we call v i1 v i3 superfluous edge. It is a redundant edge in terms of reachability. 2.2 Minimum Equivalent Graph We observe the reachability of a digraph when removing edges, until one more removing would change the reachability of some vertex. Then we get a Minimum Equivalent Graph. Following is the formal definition of MEG. D 0 = (V, E 0 ) is a minimum equivalent graph of a digraph D = (V, E) if: a. E 0 E, and b. v i Rv j in D 0 iff v i Rv j in D, and c. the cardinality of E 0 is minimum with respect to 1.&2.

6 Observation of edge removal can lead us to the following claims, which were proved to be true (Moyles). a. Removing edges in E D does not affect reachability within any of the strong components. b. Let E hk E D contains all parallel edges from V h to V k. If it is nonempty, then removing all edges except one in it does not affect the reachability of D. c. MEG of D* can be obtained by removing superfluous edges of D* in any order. d. A MEG for a strongly connected graph is a minimum complete sequence. 3.3 Algorithms to calculate MEG From the above results, we can construct the algorithm to calculate the MEG of an arbitrary digraph, as follows (Moyles). Step 1. find the strong components of D O(n 3 ) Step 2. find a MEG for each component Σ i=1 m (m-1) i Step 3. remove all but one edge in each parallel edge set of D* ΣΣm i m j Step 4. remove superfluous edges of D* Σ i=2 m-1 (m-1)! / (m-i)! The heart of the algorithm is step 2, which has exponential complexity. Moyles design this step using complete sequence. His algorithm is described in pseudo-code as follows. Start at any vertex in a strong component Set counter c = 1

7 For each existing sequence of length c Construct all possible sequence of length c+1 by adding one more vertex that is connected to the last vertex in the sequence. If one or more of the sequences include all vertices of the component, they are MEG. Exit. Else c=c+1 End For-loop Hsu found some cases under which the above algorithm could not give the MEG. He presented in his paper an algorithm using adjacency matrix, which is easier to implement. However, that cannot change the fact that the calculation of MEG of an arbitrary digraph is NP-hard. In 1995 Khuller et al. used approximation algorithm to achieve performance guarantee of 1.75 in polynomial time. Their algorithm looks for as long a cycle as it can in a strongly connected digraph and melt it, and keeps doing this until the digraph becomes one vertex. For i = k, k-1,, 2 While the graph contains a cycle of length i Melt the cycle into one vertex End while-loop End for-loop Return all the melted edges Khuller et al. studied the MEG problem of strongly connected digraphs further by bounding the problem with the length of the longest cycle in digraphs. In 1996 they concluded that MEG problem of a strongly connected digraph with maximum cycle length of 3 can be solved in polynomial time, while it is NP-hard for maximum cycle length of 5. They then applied the result to arbitrary case and improved the approximation algorithm to a performance guarantee of about 1.61.

8 3. A Potential Application of Reachability The research results on graph connectivity have had successful applications on computer network architecture. When we turn our eyes from the physical world to the virtual one, we may also be able to find many potential applications. One of the interesting possible uses coming into mind when studying graph connectivity is to analyze and design the reachability of agent hosts in a mobile agent system. An abstract mobile agent system is a large scale distributed system with only two types of entities, mobile agents and agent hosts. A mobile agent attaches to an agent host for some task at a time, taking the services of that host, and can ask the host to move it to another known host if necessary. Provided the large system scale and dynamic architecture, it is less possible and less efficient for an agent to know all hosts at the beginning. Rather, the information about hosts is distributed on hosts all over the system. It would be neat to equip an agent with enough intelligence to find out needed host on the fly. The only source a mobile agent can take advantages of is the knowledge of agent hosts that the agent travels through. A mobile agent can learn about other hosts from the current host. A good analogue in real application is search engines (Huang). So an agent host must advertise or register itself to some other hosts if it wants to provide some agent services. A host is reachable by mobile agents from all the hosts knowing this host. If we model a mobile agent system using a digraph, knowledge distribution could be designed based on the theory of graph connectivity, and host reachability in real time could be calculated based on the knowledge distribution. The best of this is, because of the formal use of graph theory, the calculation could be carried out on the fly by intelligent agents without human interference.

9 To make it more formal, we define a knowsabout relation on a set of agent hosts V = { v 1,, v n }. v i knowsabout v j means that host v i knows all the public attributes of host v j, and knows how to contact v j, for example its IP address and port number. An mobile agent at host v i can learn about the public attributes of host v j, and move to it using the contact information if it is what the agent wants. Let the set of knowsabout relationship between hosts to be K, in which k ij is an element if v i knowsabout v j. So we get a host digraph H = (V, K), where agent hosts are vertices and v i knowsabout v j is represented as a directed edge from host v i to host v j. No surprise, the reachability relation of this digraph is exactly the reachability of agent hosts. In this way, theorems of graph connectivity can be used to derive properties of existing systems, and to design new systems. For example, the following is a claim that obviously can be easily proved. Claim: if H is finite and strongly connected, then a mobile agent with enough learning ability can move from any host to any other host in finite steps. Many other properties of mobile agent systems can be studied in similar way, applying graph connectivity theory. For instance, when we include mobile agents to be vertices as well, we may define an admissibleto relation between agents and agent hosts, or a canmoveto relation which is a tuple (agent, current host, target host). Then theorems can be applied and more system properties can be reasoned about. 4. Summary In this paper we study the general graph connectivity, and specifically focus on reachability in digraphs. Algorithms for finding minimum equivalent graph of a digraph is described. Then we discuss the possible application of graph connectivity in the analysis and design of mobile agent systems.

10 5. References Hsu, Harry T.. An Algorithm for Finding a Minimal Equivalent Graph of a Digraph. Journal of the ACM, Vol.22, No.1, pp January Huang, Freeman Y. and David B. Skillicorn. The Spider Model of Agents. In Proc. of the Third International Workshop on Mobile Agents for Telecommunication Applications (MATA2001). LNCS. volume 2164, pp Montreal, Canada. August Khuller, Samir, et al.. Approximating the Minimum Equivalent Digraph. SIAM J. Computing, Vol.24, pp Khuller, Samir, et al.. On Strongly Connected Digraphs with Bounded Cycle Length. DAMATH. Vol.69, pp Moyles, Dennis M., and Gerald L. Thompson. An Algorithm for Finding a Minimum Equivalent Graph of a Digraph. Journal of the ACM, Vol.16, No.3, pp July West, Douglas Brent. Introduction to Graph Theory. Prentice Hall, Upper Saddle River, NJ, USA