On Minimum Connecting Transition Sets in Graphs

TL;DR

This paper investigates the problem of finding the smallest set of permitted transitions in a graph that allows traversal between any two vertices, proves its NP-hardness, and explores approximation algorithms and theoretical tools for solving it.

Contribution

It introduces the problem of minimum connecting transition sets, proves its NP-hardness, and develops new theoretical tools and approximation algorithms for this problem.

Findings

The problem is NP-hard.

Developed approximation algorithms.

Created theoretical tools for analysis.

Abstract

A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of transitions needed to be able to go from any vertex of the given graph to any other. We prove that this problem is NP-hard and study approximation algorithms. We develop theoretical tools that help to study this problem.