We introduce a general framework for positional games in which players score points by claiming a prescribed portion of each winning set, extending the notion of scoring Maker-Breaker games. In the scoring variant, Maker gains a point by fully claiming a winning set, while Breaker aims to minimize Maker's total score. In this paper, we generalize these models for all k-uniform positional games by fixing an integer threshold s in 1,2,..., k so that a player scores a point whenever she claims at least s elements of a winning set of size k. We refer to this class as s-of-k games. Such formulation allows for a flexible description of scoring objectives that appear in both theoretical models and real-life board games. We further investigate the impact of strategy restrictions on the achievable score. In particular, we analyze s-of-k games both under optimal play, where the score is denoted by SC, and under the additional constraint that Maker is restricted to a pairing strategy. The corresponding score in this setting is denoted by SC_2. While the unrestricted score captures the standard notion of optimal play in scoring positional games, the pairing-restricted score allows us to observe Maker's loss incurred by limiting her to these standard strategies. We comprehensively study s-of-k games played on regular grids, which provide a natural and uniform setting for illustrating the general framework. After developing several general tools for the analysis of both scores, we complement them by a number of ad-hoc strategies tailored for particular cases of these games, to obtain both upper and lower bounds for the two scores on triangular, square, rhombus and hexagonal grids.

We initiate the study of the algorithmic complexity of Maker-Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the $H$-game. Maker wins if she claims the edges of a perfect matching in the first, and a copy of a fixed graph $H$ in the second. We prove that deciding who wins the perfect matching game and the $H$-game is PSPACE-complete, even for the latter in small-diameter graphs if $H$ is a tree. Toward finding the smallest graph $H$ for which the $H$-game is PSPACE-complete, we also prove that such an $H$ of order 51 and size 57 exists. We then give several positive results for the $H$-game. As the $H$-game is already PSPACE-complete when $H$ is a tree, we mainly consider the case where $H$ belongs to a subclass of trees. In particular, we design two linear-time algorithms, both based on structural characterizations, to decide the winners of the $P_4$-game in general graphs and the $K_{1,\ell}$-game in trees. Then, we prove that the $K_{1,\ell}$-game in any graph, and the $H$-game in trees are both FPT parameterized by the length of the game, notably adding to the short list of games with this property, which is of independent interest. Another natural direction to take is to consider the $H$-game when $H$ is a cycle. While we were unable to resolve this case, we prove that the related arboricity-$k$ game is polynomial-time solvable. In particular, when $k=2$, Maker wins this game if she claims the edges of any cycle.

We study the Maker-Maker version of the domination game introduced in 2018 by Duchêne et al. Given a graph, two players alternately claim vertices. The first player to claim a dominating set of the graph wins. As the Maker-Breaker version, this game is PSPACE-complete on split and bipartite graphs. Our main result is a linear time algorithm to solve this game in forests. We also give a characterization of the cycles where the first player has a winning strategy.

The Maker–Breaker domination game is a positional game played on a graph by two players called Dominator and Staller. The players alternately select a vertex of the graph that has not yet been chosen. Dominator wins if at some point the vertices she has chosen form a dominating set of the graph. Staller wins if Dominator cannot form a dominating set. Deciding if Dominator has a winning strategy has been shown to be a PSPACE-complete problem even when restricted to chordal or bipartite graphs. In this paper, we consider strategies for Dominator based on partitions of the graph into basic subgraphs where Dominator wins as the second player. Using partitions into cycles and edges (also called perfect [1,2]-factors), we show that Dominator always wins in regular graphs and that deciding whether Dominator has a winning strategy as a second player can be computed in polynomial time for outerplanar and block graphs. We then study partitions into subgraphs with two universal vertices, which is equivalent to considering the existence of pairing dominating sets with adjacent pairs. We show that in interval graphs, Dominator wins if and only if such a partition exists. In particular, this implies that deciding whether Dominator has a winning strategy playing second is in NP for interval graphs. We finally provide an algorithm in n^k+3 for interval graphs with at most k nested intervals.

In this paper, we introduce and study a new scoring game on graphs called SMASH AND GRAB. In this game, two players, called Left and Right, take turns removing a vertex of the graph as well as all of its neighbours that become isolated by this removal. For each player and each of their turns, they score the number of vertices that were removed on their turn. The game ends when there are no more vertices remaining, and the player with the highest final score wins. We denote by Ls(G) the difference between Left and Right's final scores in G when Left starts and both players play optimally (they both aim to maximise their scores). We mainly study this parameter for different graph classes. We notably prove that Ls(F) >= 0 for any forest F (i.e., the first player cannot lose). We then use this result to compute the exact value of Ls(G) for particular forests such as unions of paths and subdivided stars. The result in paths then solves the case of a unique cycle. Finally, we prove that, for a generalisation of the game, computing the score is PSPACE-complete.

The game Influence is a scoring combinatorial game that has been introduced in 2021 by Duchêne et al. [5]. It is a good representative of Milnor's universe of scoring games, i.e. games where it is never interesting for a player to miss their turn. New general results are first given for this universe, by transposing the notions of mean and temperature derived from non-scoring combinatorial games. Such results are then applied to Influence to refine the case of unions of segments started by Duchêne et al. [5]. The computational complexity of the score of the game is also solved and proved to be PSPACE-complete. We finally focus on some specific cases of Influence when the graph is bipartite, by giving explicit strategies and bounds on the optimal score on structures like grids, hypercubes or tori.

We propose a generalization of positional games, supplementing them with a restriction on the order in which the elements of the board are allowed to be claimed. We introduce poset positional games, which are positional games with an additional structure a poset on the elements of the board. Throughout the game play, based on this poset and the set of the board elements that are claimed up to that point, we reduce the set of available moves for the player whose turn it is an element of the board can only be claimed if all the smaller elements in the poset are already claimed. We proceed to analyze these games in more detail, with a prime focus on the most studied convention, the Maker-Breaker games. First we build a general framework around poset positional games. Then, we perform a comprehensive study of the complexity of determining the game outcome, conditioned on the structure of the family of winning sets on the one side and the structure of the poset on the other.

Computer science unplugged is a scientific popularization project initiated in the 1990s by a team of New Zealand researchers. It enables participants to discover the major concepts of computer science, without a computer, through physical activies or the use of physical material, and also to initiate them into the computer science research process. This device, which is currently widely considered by both mediators and teachers, requires an in-depth analysis as a tool for transmitting knowledge. This analysis is currently in its beginning. Through three examples of popularization situations in computer science unplugged, we propose lines of thought based on observations with the perspective of a more complete didactic analysis of these situations and their transposition into classrooms.L'informatique débranchée est un dispositif de médiation scientifique introduit dans les années 1990 par une équipe de chercheurs néo-zélandais. Elle permet, sans ordinateur et via des mises en scène corporelles ou l'utilisation de matériel original, de faire découvrir aux participant.e.s les grands concepts de la science informatique, mais aussi de les initier à la démarche de recherche en informatique. Ce dispositif actuellement largement considéré à la fois par les médiateur.rice.s et les enseignant.e.s nécessite une analyse approfondie en tant qu'outil de transmission de savoir. Cette analyse n'en est actuellement qu'à ses balbutiements. A travers trois exemples de situation de médiation en informatique débranchée (SMID), nous proposons des pistes de réflexions issues d'observations sur le terrain en vue d'une analyse didactique des SMID plus complète et d'une transposition en classe.

Positional games have been introduced by Hales and Jewett in 1963 and have been extensively investigated in the literature since then. These games are played on a hypergraph where two players alternately select an unclaimed vertex of it. In the Maker-Breaker convention, if Maker manages to fully take a hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker convention, the first player to take a hyperedge wins. In both cases, the game stops as soon as Maker has taken a hyperedge. By definition, this family of games does not handle scores and cannot represent games in which players want to maximize a quantity. In this work, we introduce scoring positional games, that consist in playing on a hypergraph until all the vertices are claimed, and by defining the score as the number of hyperedges a player has fully taken. We focus here on Incidence, a scoring positional game played on a 2-uniform hypergraph, i.e. an undirected graph. In this game, two players alternately claim the vertices of a graph and score the number of edges for which they own both end vertices. In the Maker-Breaker version, Maker aims at maximizing the number of edges she owns, while Breaker aims at minimizing it. In the Maker-Maker version, both players try to take more edges than their opponent. We first give some general results on scoring positional games such that their membership in Milnor's universe and some general bounds on the score. We prove that, surprisingly, computing the score in the Maker-Breaker version of Incidence is PSPACE-complete whereas in the Maker-Maker convention, the relative score can be obtained in polynomial time. In addition, for the Maker-Breaker convention, we give a formula for the score on paths by using some equivalences due to Milnor's universe. This result implies that the score on cycles can also be computed in polynomial time.

In this paper, we introduce and study a new scoring game on graphs called smash and grab. In this game, two players, called Left and Right, take turns removing a vertex of the graph as well as all of its neighbours that become isolated by this removal. For each player and each of their turns, they score the number of vertices that were removed on their turn. The game ends when there are no more vertices remaining, and the player with the highest final score wins. We denote by Ls(G) the difference between Left and Right's final scores in G when Left starts and both players play optimally (they both aim to maximise their scores).

A k-edge colouring (not necessarily proper) of a graph with colours in { 1 , 2 , . . . , k } is neighbour sum distinguishing if, for any two adjacent vertices, the sums of the colours of the edges incident with each of them are distinct. The smallest value of k such that such a colouring of G exists is denoted by chi(e)(sigma)(G ) . When we add the additional restriction that the k-edge colouring must be proper, then the smallest value of k such that such a colouring exists is denoted by chi(sigma)'(G). Such colourings are studied on a connected graph on at least 3 vertices. There are two famous conjectures on these edge colourings: the 1-2-3 Conjecture states that chi(sigma)e & nbsp;(G ) <= 3 for any graph G ; and the other states that chi(sigma)'(G) <= delta(G) + 2 for any graph G &NOTEQUexpressionL; C-5 . In this paper, we generalize these two versions of neighbour sum distinguishing edge colourings by introducing the edge colouring in which each monochromatic set of edges induces a subgraph with maximum degree at most d. We call such an edge colouring that distinguishes adjacent vertices a neighbour sum distinguishing d-relaxed k-edge colouring. We denote by chi(sigma)'(d) (G ) the smallest value of k such that such a colouring of G exists. We study families of graphs for which chi(sigma)'& nbsp;is known. We show that the number of required colours decreases when the proper condition is relaxed. In particular, we prove that chi(sigma)'2 (G ) <= 4 for every subcubic graph. For complete graphs, we show that chi(sigma)'d (K-n) <= 4 if d is an element of { n-1/2, . . . , n -1 } and we also determine the exact value of chi(sigma)'2 (K-n). Finally, we determine the value of chi(sigma)'d (T ) for any tree T . (C)& nbsp;2021 Elsevier Inc. All rights reserved.

Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with $n \in S_L$ (resp. $n \in S_R$) when it is Left's (resp. Right's) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i.e. an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate the other kinds of behaviors for the outcome sequence. In addition to dominance, three other disjoint behaviors are defined, namely weak dominance, fairness and ultimate impartiality. We consider the problem of computing this behavior with respect to $S_L$ and $S_R$, which is connected to the well-known Frobenius coin problem. General results are given, together with arithmetic and geometric characterizations when the sets $S_L$ and $S_R$ have size at most 2.

We introduce the game INFLUENCE, a scoring combinatorial game, played on a directed graph where each vertex is either colored black or white. The two players, Black and White play alternately by taking a vertex of their color and all its successors (for Black) or all its predecessors (for White). The score of each player is the number of vertices he has taken. We prove that INFLUENCE is a nonzugzwang game, meaning that no player has interest to pass at any step of the game, and thus belongs to Milnor's universe. We study this game in the particular class of paths where black and white are alternated. We give an almost tight strategy for both players when there is one path. More precisely, we prove that the first player always gets a strictly better score than the second one, but that the difference between the score is bounded by 5. Finally, we exhibit some graphs for which the initial proportion of vertices of the color of a player is as small as possible but where this player can get almost all the vertices.

We introduce cut, the class of 2-player partition games. These are nim type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can perform on a single heap. In normal play, the player with the last move wins, and the famous Sprague–Grundy theory provides a solution. We prove that several rulesets have a periodic or an arithmetic periodic Sprague–Grundy sequence (i.e. they can be partitioned into a finite number of arithmetic progressions of the same common difference). This is achieved directly for some infinite classes of games, and moreover we develop a computational testing condition, demonstrated to solve a variety of additional games. Similar results have previously appeared for various classes of games of take-and-break, for example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and Guy (1982). In this context, our contribution consists of a systematic study of the subclass ‘break-without-take’.

We introduce the Maker-Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, select a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker-Breaker games, that is studied here in a combinatorial context. In this paper, we first prove that deciding the winner of the Maker-Breaker domination game is PSPACE-complete, even for bipartite graphs and split graphs. It is then showed that the problem is polynomial for cographs and trees. In particular, we define a strategy for Dominator that is derived from a variation of the dominating set problem, called the pairing dominating set problem.

This work is a contribution to the study of rewrite games. Positions are finite words, and the possible moves are defined by a finite number of local rewriting rules. We introduce and investigate taking-and-merging games, that is, where each rule is of the form a^k->epsilon. We give sufficient conditions for a game to be such that the losing positions (resp. the positions with a given Grundy value) form a regular language or a context-free language. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games. Finally we show that more general rewrite games quickly lead to undecidable problems. Namely, it is undecidable whether there exists a winning position in a given regular language, even if we restrict to games where each move strictly reduces the length of the current position. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games.

Six authors tell their stories from their encounters with the famous combinatorial game WYTHOFF NIM and its sequences, including a short survey on exactly covering systems. The volume of the mathematical study of this game is 59% of that of the most ubiquitous game CHESS (MathSciNet). The former originated in 1907, the latter in antiquity. Thus the mathematical study of WYTHOFF NIM may surpass that of CHESS!

We study operators that combine combinatorial games. This field was initiated by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R 1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2), we build the push-the-button game R 1 R 2 , where players start by playing according to the rules R 1 , but at some point during play, one of the players must switch the rules to R 2 , by pushing the button ". Thus, the game ends according to the terminal condition of ruleset R 2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R 1 is the game where the players put the domino tiles horizontally and R 2 the game where they play vertically (thus generalizing the octal game 0.07).

Taking-and-breaking games are combinatorial games played on heaps of tokens, where both players are allowed to remove tokens from a heap and/or split a heap into smaller heaps. Subtraction games, octal and hexadecimal games are well-known families of such games. We here consider the set of pure breaking games, that correspond to the family of taking-and-breaking games where splitting heaps only is allowed. The rules of such games are simply given by a list L of positive integers corresponding to the number of sub-heaps that a heap must be split into. Following the case of octal and hexadecimal games, we provide a computational testing condition to prove that the Grundy sequence of a given pure breaking game is arithmetic periodic. In addition, the behavior of the Grundy sequence is explicitly given for several particular values of L (e.g. when 1 is not in L or when L contains only odd values). However, despite the simplicity of its ruleset, the behavior of the Grundy function of the game having L = {1, 2} is open.