http://didawiki.di.unipi.it/lib/exe/fetch.php/magistraleinformatica/alg2/algo2_13/mfvs.pdf

On enumerating all minimal solutions of feedback problems

This paper describes an algorithm for the identification of a minimal feedback vertex set of an arbitrary directed graph. Such a set of vertices has the property that removal of these vertices from the graph eliminates all directed loops. An efficient solution to this identification problem is a familiar unsolved problem of contemporary graph theory. Although pathological examples exist for which excessive computation v2 time may be required, the described algorithm appears to be quite v efficient for large graphs arising in practical applications. The algorithm seems to have great potential in the field of test generation for sequential logic circuits and certain types of simulation algorithms. No attempt is V3 made to impose restrictive conditions on the particular minimum feedback vertex set selected, although such restrictions may at times be desirable. The algorithm has been implemented on an IBM 360/67 and has been successfully applied to practical (logic circuit) problems having in excess of 2500 vertices.

The identification of a minimal feedback vertex set of a directed graph

We study the input-output stability of arbitrary interconnections of nonlinear, time-varying, multivariable subsystems which may be either continuous-time or discrete-time. All subsystems are specified by their input-output description. We use an algorithmic decomposition of the overall system into a hierarchy of strongly connected subsystems (SCS) interacting through inter-connection subsystems (IS). Theorem I establishes that the overall system is stable once the SCS's and the IS's are stable. Theorem II shows that, under very reasonable assumptions, these sufficient conditions are actually necessary. Using the concept of minimum essential set, we partition each SCS into two parts: one part corresponds to the "forward subsystem" whose only feedbacks are self-loops; the other part corresponds to the vertices of the minimum essential set; together they form the overall feedback system. Simplified stability conditions for the SCS are obtained by exploiting this structural decomposition. Theorem III gives sufficient condition for stability of nonlinear time-varying SCS's. The other three theorems consider linear time-invariant continuous-time, lumped as well as distributed SCS's. We provide translation rules for reformulating these three theorems for the discrete-time case. We also show how to reduce the amount of calculation involved in finding the SCS characteristic polynomial.

Input-output stability theory of interconnected systems using decomposition techniques

We study the input-output stability of an arbitrary interconnection of multi-input, multi-output subsystems which may be either continuous-time or discrete-time. We consider, throughout, three types of dynamics: nonlinear time-varying, linear time-invariant distributed, and linear time-invariant lumped. First, we use the strongly connected component decomposition to aggregate the subsystems into strongly-connected-subsystems (SCS's) and interconnection-subsystems (IS's). These SCS's and IS's are then aggregated into column-subsystems (CS's) so that the overall system becomes a hierarchy of CS's. The basic structural result states that the overall system is stable if and only if every CS is stable. We then use the minimum-essential-set decomposition on each SCS so that it can be viewed as a feedback interconnection of aggregated subsystems where one of them is itself a hierarchy of subsystems. Based on this decomposition, we present results which lead to sufficient conditions for the stability of an SCS. For linear time-invariant (transfer function) dynamics, we obtain a characteristic function which gives the necessary and sufficient condition for the overall system stability. We point out the computational saving due to the decompositions in calculating this characteristic function. We believe that decomposition techniques, coupled with other techniques such as model reduction, aggregation, singular, and nonsingular perturbations, will play key roles in large scale system design.

Input-output stability of interconnected systems using decompositions: An improved formulation

A partitioning strategy of sparse matrices is dealt with. In particular, the problem of transforming a nonsingular matrix by symmetric permutation to an optimal bordered triangular form (BTF) is solved. It is shown that the problem is equivalent to the determination of a minimum essential set of a directed graph. An efficient algorithm is given for finding minimum essential sets of a digraph. The method depends on, as a preliminary step, graph simplication using local information at a vertex. A circuit-generation technique based on vertex elimination is then introduced. The algorithm is illustrated with a complete example. A simple electrical network is used to illustrate the use of the BTF in the sparse tableau approach of network analysis.

The bordered triangular matrix and minimum essential sets of a digraph

This work deals with the problem of finding minimal or quasi-minimal essential sets in directed graphs. Mainly based on a topological approach recently suggested, a modified algorithm for minimal essential sets is pointed out, and a new algorithm is proposed for quasi-minimal essential sets.

M. Courvoisier

Papers

A note on minimal and quasi-minimal essential sets in complex directed graphs