Integer programming formulations for the k$k$‐in‐a‐tree problem in graphs

Abstract

AbstractChudnovsky and Seymour proposed the Three‐in‐a‐tree algorithm which solves the following problem in polynomial time: given three fixed vertices in a simple finite graph, check whether an induced tree containing these vertices exists. In this paper, we deal with a generalization of this problem, referred to henceforth as ‐in‐a‐tree. The ‐in‐a‐tree checks whether a graph contains an induced tree spanning given vertices. When is part of the input, the problem is known to be NP‐complete. If is a fixed given number, its complexity is an open question, although there are efficient algorithms for restricted cases such as claw‐free graphs, graphs with a girth of at least and chordal graphs. We present mixed‐integer programming formulations for this problem, and we show that instances with up to 25,000 vertices can be solved in reasonable computational time.