New Applied Problems in the Theory of Acyclic Digraphs

Gurami Tsitsiashvili,Victor Bulgakov

doi:10.3390/math10010045

Gurami Tsitsiashvili, Victor Bulgakov

Open Access

https://doi.org/10.3390/math10010045

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Abstract

The following two optimization problems on acyclic digraph analysis are solved. The first of them consists of determining the minimum (in terms of volume) set of arcs, the removal of which from an acyclic digraph breaks all paths passing through a subset of its vertices. The second problem is to determine the smallest set of arcs, the introduction of which into an acyclic digraph turns it into a strongly connected one. The first problem was solved by reduction to the problem of the maximum flow and the minimum section. The second challenge was solved by calculating the minimum number of input arcs and determining the smallest set of input arcs in terms of the minimum arc coverage of an acyclic digraph. The solution of these problems extends to an arbitrary digraph by isolating the components of cyclic equivalence in it and the arcs between them.

Highlights

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An alternative and in some sense inverse second problem is connected with the introduction into the digraph of a smallest set of new arcs that turn an acyclic digraph into a strongly connected one
This procedure is needed to include all the vertices of the acyclic digraph in the feedbacks that stabilize the functioning of the network represented by the digraph

Summary

Introduction

An alternative and in some sense inverse second problem is connected with the introduction into the digraph of a smallest set of new arcs that turn an acyclic digraph into a strongly connected one (in which there is a path from any vertex to any other vertex) This procedure is needed to include all the vertices of the acyclic digraph in the feedbacks that stabilize the functioning of the network represented by the digraph. The paper presents an original algorithm for solving the problem of allocating cyclic equivalence classes [29] basing on the sequential inclusion in the digraph of a new vertex and arcs connecting it to the already specified ones

Optimal Blocking of Selected Vertices of the Acyclic Digraph

Recurrent Algorithm for Class Allocation Cyclic Equivalence

Findings

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