Assistance and Interdiction Problems on Interval Graphs

TL;DR

This paper introduces a new framework for graph modifications on interval graphs, studying interdiction and assistance problems related to key graph parameters, and provides polynomial-time algorithms for many of these problems.

Contribution

It presents a novel framework for interval graph modifications and analyzes interdiction and assistance problems, offering polynomial algorithms for several key graph parameters.

Findings

Polynomial algorithms for most interdiction and assistance problems.

Most vital nodes problem can be solved in polynomial time on interval graphs.

Framework applies to parameters like independence number, clique number, shortest paths, and scattering number.

Abstract

We introduce a novel framework of graph modifications specific to interval graphs. We study interdiction problems with respect to these graph modifications. Given a list of original intervals, each interval has a replacement interval such that either the replacement contains the original, or the original contains the replacement. The interdictor is allowed to replace up to $k$ original intervals with their replacements. Using this framework we also study the contrary of interdiction problems which we call assistance problems. We study these problems for the independence number, the clique number, shortest paths, and the scattering number. We obtain polynomial time algorithms for most of the studied problems. Via easy reductions, it follows that on interval graphs, the most vital nodes problem with respect to shortest path, independence number and Hamiltonicity can be solved in polynomial time.