A Survey on the Complexity of Flood-Filling Games

Abstract

This survey is offered in honour of the special occasion of the birthday celebration of science and education pioneer Professor Juraj Hromkovic. In this survey, we review recent results on one-player flood-filling games on graphs, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. As for many colored graph problems, flood-filling games have relevant interpretations in bioinformatics. The original versions of Flood-It and Free-Flood-It are played on \(n \times m\) grids, but several studies were devoted to analyzing the complexity of these games when the “board” (the graph) belongs to other graph classes. A complete mapping of the complexity of flood-filling games on trees is presented, charting the consequences of single and aggregate parameterizations. The Flood-It problem on trees and the Restricted Shortest Common Supersequence (RSCS) problem are analogous. Flood-It remains NP-hard when played on 3-colored trees. A general framework for reducibility from Flood-It to Free-Flood-It is revisited. The complexity behavior of these games when played on various kinds of graphs is surveyed, such as Cartesian products of cycles and paths, circular grids, split graphs, co-comparability graphs, and AT-free graphs. We review a recent investigation of the parameterized complexity of Flood-It when the size of a minimum vertex cover is the structural parameter. Some educational aspects of the game are also reviewed. Happy Birthday, Juraj!