arXiv subjects

Valentin Gledel

Publications and source records attributed to Valentin Gledel.

At least 19 recordsLinked to original sources

arXiv2026

On the damage number of graphs

We study a variant of Cops and Robbers in which the robber attempts to visit as many vertices of the graph as possible without being captured, while the cop aims to keep the robber confined to a small set of vertices. The \textit{damage number} of a graph $G$, introduced by Cox and Sanaei in 2019, is the maximum number of vertices the robber can visit in a game of Cops and Robbers on $G$. In this paper, we determine damage numbers for several classes of graphs, including hypercubes, Hamming graphs, Johnson graphs, incidence graphs of projective planes, and Erd\H{o}s-R\'enyi random graph $G(n,p)$, for $p \gg \log^{2/5}(n) / n^{1/5}$. We also show that the problem of determining the damage number of a graph is {\sf PSPACE}-complete.

math.CO↗

arXiv2026

Positional s-of-k games

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.

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arXiv2025

The Avoider-Enforcer game on hypergraphs of rank 3

In the Avoider-Enforcer convention of positional games, two players, Avoider and Enforcer, take turns selecting vertices from a hypergraph H. Enforcer wins if, by the time all vertices of H have been selected, Avoider has completely filled an edge of H with her vertices; otherwise, Avoider wins. In this paper, we first give some general results, in particular regarding the outcome of the game and disjoint unions of hypergraphs. We then determine which player has a winning strategy for all hypergraphs of rank 2, and for linear hypergraphs of rank 3 when Avoider plays the last move. The structural characterisations we obtain yield polynomial-time algorithms.

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arXiv2024

On the Metric Dimension of $K_a \times K_b \times K_c$

In this work we determine the metric dimension of $ K_a \times K_b \times K_c$ for all $a,b,c\in \mathbb N$ with $ a \le b \le c $ as follows. For $3a c$, it is $\left \lfloor \frac{2}{3}(b+c-1) \right \rfloor$, and for $3a=b+c$, it is $\left \lfloor \frac{a+b+c}{2} \right \rfloor -1 $. The only open case is $3a>b+c$, where two values are possible, namely $\left \lfloor \frac{a+b+c}{2} \right \rfloor -1 $ and $\left \lfloor \frac{a+b+c}{2} \right \rfloor $. This result extends previous results of C\'acere et al., who computed the metric dimension of $ K_a \times K_b$, and of Drewes and J\"ager, who computed the metric dimension of $ K_a \times K_a \times K_a$. We prove our result by introducing and analyzing a new variant of Static Black-Peg Mastermind, in which each peg has its own permitted set of colors. For all cases, we present strategies which we prove to be both feasible and optimal. Our main result follows, as the number of questions of these strategies is equal to the metric dimension of $K_a \times K_b \times K_c$.

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arXiv2024

On the complexity of Client-Waiter and Waiter-Client games

Positional games were introduced by Hales and Jewett in 1963, and their study became more popular after Erdos and Selfridge's first result on their connection to Ramsey theory and hypergraph coloring in 1973. Several conventions of these games exist, and the most popular one, Maker-Breaker was proved to be PSPACE-complete by Schaefer in 1978. The study of their complexity then stopped for decades, until 2017 when Bonnet, Jamain, and Saffidine proved that Maker-Breaker is W[1]-complete when parameterized by the number of moves. The study was then intensified when Rahman and Watson improved Schaefer's result in 2021 by proving that the PSPACE-hardness holds for 6-uniform hypergraphs. More recently, Galliot, Gravier, and Sivignon proved that computing the winner on rank 3 hypergraphs is in P. We focus here on the Client-Waiter and the Waiter-Client conventions. Both were proved to be NP-hard by Csernenszky, Martin, and Pluh\'ar in 2011, but neither completeness nor positive results were known for these conventions. In this paper, we complete the study of these conventions by proving that the former is PSPACE-complete, even restricted to 6-uniform hypergraphs, and by providing an FPT-algorithm for the latter, parameterized by the size of its largest edge. In particular, the winner of Waiter-Client can be computed in polynomial time in k-uniform hypergraphs for any fixed integer k. Finally, in search of finding the exact bound between the polynomial result and the hardness result, we focused on the complexity of rank 3 hypergraphs in the Client-Waiter convention. We provide an algorithm that runs in polynomial time with an oracle in NP.

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arXiv2024

Partition strategies for the Maker-Breaker domination game

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 $k$-nested interval graphs (i.e. interval graphs with at most $k$ intervals included one in each other).

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arXiv2024

Simple Chopsticks: Playing with any number of hands and fingers

Chopsticks is a game played by two players where they start with one finger raised on each hand. On their turn, each player moves by pointing an attacking hand at one of their opponent's hands. The number of fingers on the pointed hand increases by the number of fingers on the attacking hand. If, after a move, a hand contains more than five fingers, it is removed from play. There are also other rules that allow players to move fingers from one hand to another, but we focus on this simple setup. We introduce a generalization of Chopsticks, called Simple Chopsticks, in which the players may have any number of $n$-fingered hands. We find that having more hands than your opponent is generally good, and use this fact to fully characterize the outcomes of \octopus/ in the case where the players have 2-fingered hands.

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arXiv2024

Poset Positional Games

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 analyse 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.

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arXiv2023

A note on the flip distance between non-crossing spanning trees

We consider spanning trees of $n$ points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree. Given two trees, we investigate the minimum number of flips required to transform one into the other. The naive $2n-\Omega(1)$ upper bound stood for 25 years until a recent breakthrough from Aichholzer et al. yielding a $2n-\Omega(\log n)$ bound. We improve their result with a $2n-\Omega(\sqrt{n})$ upper bound, and we strengthen and shorten the proofs of several of their results.

cs.CG↗

arXiv2023

Complexity of Maker-Breaker Games on Edge Sets of Graphs

We study 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.

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arXiv2022

Incidence, a Scoring Positional Game on Graphs

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.

math.CO↗

arXiv2022

Avoidance games are PSPACE-Complete

Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games are studied since the introduction of the game of SIM in 1968, but only few complexity results are known on them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojakovi\'c proved that these games are NP-hard. As these games corresponds to the mis\`ere version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question remained open since then. We prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete, and as a consequence of it that some particular Avoider-Enforcer games also are.

math.CO↗

arXiv2019

Maker-Breaker total domination game

Maker-Breaker total domination game in graphs is introduced as a natural counterpart to the Maker-Breaker domination game recently studied by Duch\^ene, Gledel, Parreau, and Renault. Both games are instances of the combinatorial Maker-Breaker games. The Maker-Breaker total domination game is played on a graph $G$ by two players who alternately take turns choosing vertices of $G$. The first player, Dominator, selects a vertex in order to totally dominate $G$ while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. It is shown that there are infinitely many connected cubic graphs in which Staller wins and that no minimum degree condition is sufficient to guarantee that Dominator wins when Staller starts the game. An amalgamation lemma is established and used to determine the outcome of the game played on grids. Cacti are also classified with respect to the outcome of the game. A connection between the game and hypergraphs is established. It is proved that the game is PSPACE-complete on split and bipartite graphs. Several problems and questions are also posed.

math.CO↗

arXiv2018

Maker-Breaker domination number

The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $\gamma_{{\rm MB}}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $\gamma_{{\rm MB}}'(G)$. Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant $\gamma_{{\rm MB}}(G)$ is also compared with the domination number. Using the Erd\H{o}s-Selfridge Criterion a large class of graphs $G$ is found for which $\gamma_{{\rm MB}}(G) > \gamma(G)$ holds. Residual graphs are introduced and used to bound/determine $\gamma_{{\rm MB}}(G)$ and $\gamma_{{\rm MB}}'(G)$. Using residual graphs, $\gamma_{{\rm MB}}(T)$ and $\gamma_{{\rm MB}}'(T)$ are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.

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arXiv2018

Strong geodetic number of complete bipartite graphs, crown graphs and hypercubes

The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the strong geodetic number of complete bipartite graphs is continued. The formula for $\text{sg}(K_{n,m})$ is given, as well as a formula for the crown graphs $S_n^0$. Bounds on $\text{sg}(Q_n)$ are also discussed.

math.CO↗

arXiv2018

Maker-Breaker domination game

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.

cs.DM↗

arXiv2018

Strong geodetic cores and Cartesian product graphs

The strong geodetic problem on a graph $G$ is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of $G$ are covered. To do this as efficiently as possible, strong geodetic cores and related numbers are introduced. Sharp upper and lower bounds on the strong geodetic core number are proved. Using the strong geodetic core number an earlier upper bound on the strong geodetic number of Cartesian products is improved. It is also proved that ${\rm sg}(G \,\square\, K_2) \geq {\rm sg}(G)$ holds for different families of graphs, a result conjectured to be true in general. Counterexamples are constructed demonstrating that the conjecture does not hold in general.

math.CO↗

arXiv2017

A generalization of Arc-Kayles

The game Arc-Kayles is played on an undirected graph with two players taking turns deleting an edge and its endpoints from the graph. We study a generalization of this game, Weighted Arc Kayles (WAK for short), played on graphs with counters on the vertices. The two players alternate choosing an edge and removing one counter on both endpoints. An edge can no longer be selected if any of its endpoints has no counter left. The last player to play a move wins. We give a winning strategy for WAK on trees of depth 2. Moreover, we show that the Grundy values of WAK and Arc-Kayles are unbounded. We also prove a periodicity result on the outcome of WAK when the number of counters is fixed for all the vertices but one. Finally, we show links between this game and a variation of the non-attacking queens game on a chessboard.

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