56 research outputs found
Minimal dominating sets enumeration with FPT-delay parameterized by the degeneracy and maximum degree
At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered
generation that yields an -delay algorithm listing all minimal
transversals of an -vertex hypergraph of degeneracy . Recently at IWOCA
2019, Conte, Kant\'e, Marino, and Uno asked whether this XP-delay algorithm
parameterized by could be made FPT-delay parameterized by and the
maximum degree , i.e., an algorithm with delay for some computable function . Moreover, as a first step toward
answering that question, they note that the same delay is open for the
intimately related problem of listing all minimal dominating sets in graphs. In
this paper, we answer the latter question in the affirmative.Comment: 18 pages, 2 figure
On a vertex-capturing game
In this paper, we study the recently introduced scoring game played on graphs called the Edge-Balanced Index Game. This game is played on a graph by two players, Alice and Bob, who take turns colouring an uncoloured edge of the graph. Alice plays first and colours edges red, while Bob colours edges blue. The game ends once all the edges have been coloured. A player captures a vertex if more than half of its incident edges are coloured by that player, and the player that captures the most vertices wins.
Using classical arguments from the field, we first prove general properties of this game. Namely, we prove that there is no graph in which Bob can win (if Alice plays optimally), while Alice can never capture more than 2 more vertices than Bob (if Bob plays optimally). Through dedicated arguments, we then investigate more specific properties of the game, and focus on its outcome when played in particular graph classes. Specifically, we determine the outcome of the game in paths, cycles, complete bipartite graphs, and Cartesian grids, and give partial results for trees and complete graphs
Non-Clashing Teaching Maps for Balls in Graphs
Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023]
introduced non-clashing teaching and showed it to be the most efficient machine
teaching model satisfying the benchmark for collusion-avoidance set by Goldman
and Mathias. A teaching map for a concept class assigns a
(teaching) set of examples to each concept . A teaching
map is non-clashing if no pair of concepts are consistent with the union of
their teaching sets. The size of a non-clashing teaching map (NCTM) is the
maximum size of a , . The non-clashing teaching dimension
NCTD of is the minimum size of an NCTM for .
NCTM and NCTD are defined analogously, except the teacher may
only use positive examples.
We study NCTMs and NCTMs for the concept class
consisting of all balls of a graph . We show that the associated decision
problem {\sc B-NCTD} for NCTD is NP-complete in split, co-bipartite,
and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails,
{\sc B-NCTD} does not admit an algorithm running in time
, nor a kernelization algorithm outputting a
kernel with vertices, where vc is the vertex cover number of .
These are extremely rare results: it is only the second (fourth, resp.) problem
in NP to admit a double-exponential lower bound parameterized by vc (treewidth,
resp.), and only one of very few problems to admit an ETH-based conditional
lower bound on the number of vertices in a kernel. We complement these lower
bounds with matching upper bounds. For trees, interval graphs, cycles, and
trees of cycles, we derive NCTMs or NCTMs for of size
proportional to its VC-dimension. For Gromov-hyperbolic graphs, we design an
approximate NCTM for of size 2.Comment: Shortened abstract due to character limi
Metric Dimension: from Graphs to Oriented Graphs
International audienceThe metric dimension of an undirected graph is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of . Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic -node graphs, all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension . We then consider strongly-connected orientations of grids. For a torus with rows and columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically (the equality holding when , are even, which is best possible). For a grid with rows and columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension
Dimension Métrique des Graphes Orientés
International audienceLa dimension métrique MD(G) d'un graphe non-dirigé G est le nombre minimum de sommets qui permettent, via leurs distances à tous les sommets, de distinguer les sommets de G les uns des autres. Cette notion a été beaucoup étudiée depuis sa conception dans les années 70 car elle permet notamment de modéliser la localisation d'une cible par ses distances à un réseau de capteurs dans un graphe. Nous considérons ici sa généralisation aux digraphes. Nous étudions, pour certaines classes de graphes, la dimension métrique maximum parmi toutes les orientations fortement connexes en donnant des bornes sur cette valeur. Notamment, nous étudions ce paramètre dans les graphes de degré maximum borné, les grilles et les tores. Pour ces derniers, nous trouvons la valeur exacte asymptotiquement
Metric Dimension: from Graphs to Oriented Graphs
International audienceThe metric dimension of an undirected graph is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of . Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic -node graphs, all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension . We then consider strongly-connected orientations of grids. For a torus with rows and columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically (the equality holding when , are even, which is best possible). For a grid with rows and columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension
The Largest Connected Subgraph Game
This paper introduces the largest connected subgraph game played on an undirected graph . In each round, Alice first colours an uncoloured vertex of red, and then, Bob colours an uncoloured vertex of blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than the order of any blue (red, resp.) connected subgraph. We first prove that Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or has at least vertices, and that Alice wins in cycles if and if only if the cycle is of odd length. Lastly, we give an algorithm to determine the outcome of the game in cographs in linear time
Connexions ! Le jeu du plus grand sous-graphe connexe
International audienceThis paper introduces the largest connected subgraph game played on an undirected graph G.In each round, Alice first colours an uncoloured vertex of G red, and then, Bob colours an uncoloured vertex of G blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than the order of any blue (red, resp.) connected subgraph. We first prove that Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or has at least 11 vertices, and that Alice wins in cycles if and only if the cycle is of odd length. Lastly, we give an algorithm to determine the outcome of the game in cographs in linear time.Nous définissons le jeu du plus grand sous-graphe connexe. Soit un graphe dont les sommets sont initialement non colorés. Tour-à-tour, le premier joueur, Alice, colore en rouge un sommet non coloré, puis le second joueur, Bob, colore un sommet non coloré en bleu, et ainsi de suite. Le jeu s'achève lorsque tous les sommets du graphe ont été colorés. Le vainqueur est le joueur dont le sous-graphe coloré a la plus grande composante connexe. Nous prouvons que, si Alice joue optimalement, Bob ne peut jamais gagner, et définissons une classe de graphes infinie, appelés graphes miroirs, dans lesquels Bob peut forcer une égalité. Du point de vue complexité, nous montrons ensuite que déterminer l'issue du jeu est PSPACE-complet même lorsque restreint aux graphes bipartis de petit diamètre, et que reconnaître un graphe miroir est GI-difficile. Enfin, nous caractérisons les chemins et cycles dans lesquels Alice gagne et nous prouvons que l’issue du jeu peut être déterminée en temps linéaire dans la classe des cographes
The Orthogonal Colouring Game
International audienceWe introduce the Orthogonal Colouring Game, in which two players alternately colour vertices (from a choice of m ∈ N colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the colouring. Each player aims to maximise her score, which is the number of coloured vertices in the copy of the graph she owns. The main result of this paper is that the second player has a strategy to force a draw in this game for any m ∈ N for graphs that admit a strictly matched involution. An involution σ of a graph G is strictly matched if its fixed point set induces a clique and any non-fixed point v ∈ V (G) is connected with its image σ(v) by an edge. We give a structural characterisation of graphs admitting a strictly matched involution and bounds for the number of such graphs. Examples of such graphs are the graphs associated with Latin squares and sudoku squares
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