A Gaussian Elimination Algorithm for the Enumeration of Cut Sets in a Graph

Abstract

By defining a suitable algebra for cut sets, it is possible to reduce the problem of enumerating the cut sets between all pairs of nodes in a graph to the problem of solving a system of linear equations An algorithm for solving thin system using Gausman elimination is presented in thin paper The efficiency of the algorithm depends on the implementation of sum and multiplication Therefore~ some properties of cut sets are investigated, which greatly slmphfy the implementation of these operations for the case of undirected graphs. The time required by the algorithm is shown to be hnear with the number of cut sets for complete graphs Some experimental results are given, proving that the efficiency of the algorithm increases by increasing the number of pairs of nodes for which the cut sets are computed.KEY WORDS AND PHRASES: graph theory, cut set enumeration, regular algebra, Gaussian elimination CR CATEGORIES, 5 20, 5 32 IntroductwnThe problem of enumerating all cut sets between all pairs of nodes in a graph is an, important combinatorial problem. For example, it can be the first step for the computation of the terminal rehabihty in a communication network [1,2,3].Several methods have been presented for enumerating all cut sets between two given nodes i and j, For instance, the method described in [2] first computes all simple paths and then finds all combinations of arcs cutting all paths. Another method [3] searches the given graph starting from node i and constructs a tree whose terminal nodes are the cut sets. Furthermore, the cut sets could be obtained by taking all hnear combinations of the cut sets belonging to a suitably constructed set [4].A regular algebra method was presented by the author in [5]. By giving a suitable algebra for cut sets, it is possible to reduce the problem of enumerating all cut sets to the problem of solving a system of linear equations in this algebra. Using this method, it is possible to compute simultaneously the cut sets between all pairs of nodes. A simalar approach has been used for enumerating all simple paths in a graph [6] or for finding shortest paths [7].]n Section 2 we present our method by giving the algebra for cut sets and by showing how our problem becomes one of solving a system of linear equations. Moreover, we give a Gaussian elimination algorithm for solving this system.A simple implementation of sum and multiplication m our algebra would make the Gaussian elimination algorithm very inefficient. Therefore, m Sections 3 and 4 we show how these operations can be implemented in an efficient way for undirected graphs by taking into account some properties of cut sets. Some aspects regarding the computa-

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“…(D p , ⊕, ⊗, {∅}, ∅) is an idempotent commutative semiring where {∅} is the additive identity and ∅ is the multiplicative identity. This is known as Martelli's semiring, originally used to compute cutsets of a graph [11,12].…”

“…Proof. Straightforward, previously given by Martelli [11]. □ In order to transform the program during adornment we need a consistent reordering function.…”

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