Toward Quantum Combinatorial Games

In this paper, we propose a Quantum variation of combinatorial games, generalizing the Quantum Tic-Tac-Toe proposed by Allan Goff. A combinatorial game is a two-player game with no chance and no hidden information, such as Go or Chess. In this paper, we consider the possibility of playing superpositions of moves in such games. We propose different rulesets depending on when superposed moves should be played, and prove that all these rulesets may lead similar games to different outcomes. We then consider Quantum variations of the game of Nim. We conclude with some discussion on the relative interest of the different rulesets

Power domination in maximal planar graphs

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n $\ge$ 6 admits a power dominating set of size at most (n--2)/4

The domination game played on unions of graphs

Abstract In a graph G, a vertex is said to dominate itself and its neighbors. The Domination game is a two player game played on a finite graph. Players alternate turns in choosing a vertex that dominates at least one new vertex. The game ends when no move is possible, that is when the set of chosen vertices forms a dominating set of the graph. One player (Dominator) aims to minimize the size of this set while the other (Staller) tries to maximize it. The game domination number, denoted by γg, is the number of moves when both players play optimally and Dominator starts. The Staller-start game domination number γ g is defined similarly when Staller starts. It is known that the difference between these two values is at most one We first describe a family of graphs that we call no-minus graphs, for which no player gets advantage in passing a move. While it is known that forests are no-minus, we prove that tri-split graphs and dually chordal graphs also are no-minus. Then, we show that the domination game parameters of the union of two no-minus graphs can take only two values according to the domination game parameters of the initial graphs. In comparison, we also show that in the general case, up to four values may be possible

Searching Posets: from dichotomy to the golden ratio

International audienceSearching a sorted array is a very classical textbook problem. In somespecial circumstances, such as searching a regression in a version controlsystem, it is necessary to search in a partially ordered set. The problemcan then be seen as trying to identify a faulty vertex in a directed acyclicgraph where each step consists in querying a vertex and asking whetherthe searched vertex is an ancestor of the current vertex. Though theproblem is known to be NP-complete in general, we put more focuson the more interesting setting when the indegree of each vertex isbounded, and in particular when it is bounded by two. We show thatwith the degree 2 restrictions, all directed acyclic graphs on n verticescan be searched in no more than logφ(n) + 1 queries, where φ standsfor the golden ratio. On the other hand, we also show that there existsdirected trees which require at least ⌈logφ(n)⌉ − 2 queries. Expectedly,the later proof expectedly involves Fibonnaci related trees. We will alsodiscuss how the NP-completeness proof of the problem can be extendedto this setting.Based on joint work with Julien Courtiel and Romain Leco