Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore.We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, playing an octal game on a path P n is equivalent to playing the same octal game on a heap of n counters.We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph. We study this game on trees and give a complete resolution of this game on subdivided stars and bistars. defined as 0.33. A precise definition will be given in Section 2. Octal games have been extensively studied. One of the most important questions [4] is the periodicity of these games. Indeed, it seems that all finite octal games have a periodic behaviour in the following sense: the set of initial numbers of counters for which the first player has a winning strategy is ultimately periodic. This is true for all subtraction games and for all finite octal games for which the study has been completed [5,1].Octal games can also be played by placing counters in a row. Heaps are constituted by consecutive counters and only consecutive counters can be removed. According to this representation, it seems natural to play octal games on more complex structures like graphs. A position of the game is a graph and players remove vertices that induce a connected component which corresponds to consecutive counters. The idea to extend the notion of octal games to graphs was already suggested in [6]. However, to our knowledge, this idea has not been further developed. With our definition, playing the generalization of an octal game on a path is the same as playing the original octal game. In the special case of subtraction games, players have to keep the graph connected. As an example, playing 0.33 on a graph consists in removing one vertex or two adjacent vertices from the graph without disconnecting it.This extension of octal games is in line with several take-away games on graphs such as Arc Kayles [7] and Grim [8]. However, it does not describe some other deletion games, such as the vertex and edge versions of the game geography [7,9], vertex and edge deletion games with parity rules, considered in [10] and [11], or scoring deletion games such as Le Pic arête [12].We will first give in Section 2 basic definitions from combinatorial game theory as well as a formal definition of octal games on graphs. We then focus on the game 0.33 which is one of the simplest octal games, and to its study on trees. We first study subdivided stars in Section 3. We prove that paths can be reduced modulo 3 which leads to a complete resolution, in contrast with the related studies on subdivided stars of Node Kayles [6] and Arc Kayles [13]. In Section 4, we extend our results to subdivide...

In the Firefighter problem, introduced by Hartnell in 1995, a fire spreads through a graph while a player chooses which vertices to protect in order to contain it. In this paper, we focus on the case of trees and we consider as well the Fractional Firefighter game where the amount of protection allocated to a vertex lies between 0 and 1. We introduce the online version of both Firefighter and Fractional Firefighter, in which the number of firefighters available at each turn is revealed over time. We show that the greedy algorithm on finite trees, which maximises at each turn the amount of vertices protected, is 1/2-competitive for both online versions; this was previously known only in special cases of Firefighter. We also show that, for Firefighter, the optimal competitive ratio of online algorithms ranges between 1/2 and the inverse of the golden ratio. The greedy algorithm is optimal if the number of firefighters is not bounded and we propose an optimal online algorithm which reaches the inverse of the golden ratio if at most 2 firefighters are available. Finally, we show that on infinite trees with linear growth, any firefighter sequence stronger than a non-zero periodic sequence is sufficient to contain the fire, even when revealed online. * corresponding author. † We acknowledge the support of GEO-SAFE, H2020-MSCA-RISE-2015 project # 691161.

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