The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

TL;DR

This paper explores the computational complexity of connectivity problems in forbidden-transition and edge-colored graphs, revealing both hardness results and tractability under certain parameters, with implications for various applied fields.

Contribution

It introduces the parameterized complexity analysis of connectivity problems in forbidden-transition graphs and identifies conditions for tractability and hardness.

Findings

Finding a compatible path is W[1]-hard when parameterized by vertex-deletion distance to a linear forest.

Properly colored Hamiltonian cycle can be found efficiently when parameterized by treewidth.

The study connects forbidden-transition graphs to practical applications in networks and bioinformatics.

Abstract

The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. We initiate the study of fundamental connectivity problems from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth of finding a properly colored Hamiltonian cycle in an edge-colored graph; properly colored walks in edge-colored graphs is one of the most studied special cases of compatible walks in forbidden-transition graphs.