2-FREE-FLOOD-IT is polynomial

We study a discrete diffusion process introduced in some combinatorial games called FLOODIT and MADVIRUS that can be played online [1, 2] and whose computational complexity has been recently studied [3, 4]. The flooding dynamics used in those games can be defined for any colored graph. It is shown in [5] that studying this dynamics directly on general graph is a valuable approach to understand its specificities and extract uncluttered key patterns or algorithms that can be applied with success to particular cases like the square grid of FLOODIT or the hexagonal grid of MADVIRUS, and many other classes of graphs. This report is the translation from french to english of the section in [5] showing that the variant of the problem called 2-Free-Flood-It can be solved with a polynomial algorithm, answering a question raised in [3, 4]. 1 Definitions and notation Let G = (V,E) be a connected undirected graph, with vertices V and edges E. The number of vertices (resp. edges) will be denoted n (resp. m). A coloration of G is a mapping from V into a set of colors C. It will be called a c-coloration if |C| = c. It will be called a proper coloration if adjacent vertices have different colors. Once a coloration of G is given, a zone Z is defined as a connected monochromatic subset of V . The dynamics which is studied consists in applying a sequence of flooding operations to an initial colored graph. In the FREE-FLOOD-IT version [3, 4] that is studied here, a flooding operation consists in choosing a zone Z and a color c and then replacing the color of all vertices in Z by c. It yields a new coloration of the initial graph where the zone Z may extend if some adjacent zones were colored by c. The game associated to this dynamics takes a colored graph as input and aims at finding the shortest length of a sequence leading to a monochromatic graph. At that time, we say that we have flooded the whole graph. Finding this shortest length is an optimization problem, which is hard for general graphs when working with 3 or more colors (NP-hardness results in [3, 4]). But for 2 colors, the optimization problem that will be called 2-Free-Flood-It has a computational complexity which remained open (the question was raised in [3, 4] for the class of square