Detecting cycles in a state space is a key task in verification algortihms like LTL/CTL model checking and, less well known, reduction modulo branching bisimulation. This paper focuses on the problem of finding cycles (strongly connected components) in very large distributed state spaces. We present a collection of state space transformations meant as building blocks for custom algorithms. We a...

For a strongly connected Extended-Free-Choice Net (EFC-net) N the algorithm given in this paper either nds a cover of S-components or recognizes that N is not well-formed. Well-formedness denotes the existence of a live and bounded marking for N. The algorithm is also adaptable to compute a cover of T-components. The time complexity is O(jPjjTj). According to the Rank Theorem, nding a cover of ...

by JIANPING ZHU (Under the Direction of Robert W. Robinson) ABSTRACT In this thesis, we consider the following problem: Given a strongly connected digraph G = (V, E), where V is the set of vertices and E is the set of edges , “is it a minimal strong connected digraph?”. A reducible edges e is one for which G−e is strongly connected. A minimal strongly connected digraph is one with no reducible ...

We present an Isabelle/HOL formalization of Gabow’s algorithm for finding the strongly connected components of a directed graph. Using data refinement techniques, we extract efficient code that performs comparable to a reference implementation in Java. Our style of formalization allows for reusing large parts of the proofs when defining variants of the algorithm. We demonstrate this by verifyin...

This paper shows how a general technique, called lock-step search, used in dynamic graph algorithms, can be used to improve the running time of two problems arising in program veriication and communication protocol design. (1) We consider the nonemptiness problem for Streett automata: We are given a directed graph G = (V; E) with n = jV j and m = jEj, and a collection of pairs of subsets of ver...

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥, we characterize strongly connected n-partite tournaments ...

The inverse Gröbner basis problem is to find the ideals that have a given monomial ideal as its initial ideal. We consider the problem of finding when the given monomial ideal is the initial ideal of a prime ideal. Kalkbrenner and Sturmfels (1995), in Theorem 1, prove that the radical of the initial ideal of a prime ideal is equi-dimensional and connected in codimension one or equivalently, the...

In Euclidean geometry we find three types of special conics, which are distinguished with respect to the Euclidean similarity group: circles, parabolas, and equilateral hyperbolas. They have on one hand special elementary geometric properties (c.f. [7]) and on the other they are strongly connected to the “absolute elliptic involution” in the ideal line of the projectively enclosed Euclidean pla...

In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribu...

This note extends the results of Frank, Jordán, and Szigeti [1] on parity constrained orientations with connectivity requirements. Given a hypergraph H, a non-negative intersecting supermodular set function p, and a preferred in-degree parity for every node, a formula is given on the minimum number of nodes with wrong in-degree parity in an orientation of H covering p. It is shown that the mini...