# Chordless paths through three vertices

## Abstract

Consider the following problem, which we call "Chordless path through three vertices" or CP3V, for short: Given a simple undirected graph G=(V,E), a positive integer k, and three distinct vertices s, t, and v∈V, is there a chordless path of length at most k from s via v to t in G? In a chordless path, no two vertices are connected by an edge that is not in the path. Alternatively, one could say that the subgraph induced by the vertex set of the path in G is the path itself. The problem has arisen in the context of service deployment in communication networks. We resolve the parametric complexity of CP3V by proving it W[1]-complete with respect to its natural parameter k. Our reduction extends to a number of related problems about chordless paths and cycles. In particular, deciding on the existence of a single directed chordless (s,t)-path in a digraph is also W[1]-complete with respect to the length of the path.