Papers by Sylvain Gravier
Ars Combinatoria -Waterloo then Winnipeg-
In a paper of Cockayne et al., the authors establish an upper and a lower bound for the dominatin... more In a paper of Cockayne et al., the authors establish an upper and a lower bound for the dominating number of the complete grid graph Gn,n of order n2. Namely, they proved a "formula", and cited two questions of Paul Erdos. One of these questions was "Can we improve the order of the difference between lower and upper bounds from n/5 to n/2 ?". Our aim here is to give a positive answer to this question.
The fiber product of graphs over homomorphims, a notion introduced by P. Hell (1972) is used here... more The fiber product of graphs over homomorphims, a notion introduced by P. Hell (1972) is used here as a new approach to S. Hedetniemi's conjecture (1966) since it's a subgraph of the cross product. We consider here only the fiber product over color-ings which are special cases of homomorphisms of graphs. We show that Khelladi's conjecture (1991): "The fiber product over colorings of two n-chromatic graphs is also an n-chromatic graph" implying trivially Hedetniemi's is true for n ≤ 3 . Moreover, we propose an equivalent statement of Khelladi's conjecture using the class of graph colourings of a graph defined by El-Zahar and Sauer.
Discrete mathematics & theoretical computer science DMTCS
Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is... more Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well.
Theoretical Computer Science, 2015
A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a dist... more A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a distinct set of vertices C such that every vertex in B has an odd number of neighbors in C. We point out the connections of weak odd domination with odd domination, [σ, ρ]-domination, and perfect codes. We introduce bounds on κ(G), the maximum size of WOD sets of a graph G, and on κ ′ (G), the minimum size of non WOD sets of G. Moreover, we prove that the corresponding decision problems are NP-complete. The study of weak odd domination is mainly motivated by the design of graphbased quantum secret sharing protocols : a graph G of order n corresponds to a secret sharing protocol which threshold is κQ(G) = max(κ(G), n − κ ′ (G)). These graph-based protocols are very promising in terms of physical implementation, however all such graph-based protocols studied in the literature have quasi-unanimity thresholds (i.e. κQ(G) = n − o(n) where n is the order of the graph G underlying the protocol). In this paper, we show using probabilistic methods, the existence of graphs with smaller κQ (i.e. κQ(G) ≤ 0.811n where n is the order of G). We also prove that deciding for a given graph G whether κQ(G) ≤ k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a κQ smaller than 0.811n.
The electronic journal of combinatorics
We determine the minimum cardinality of an identifying code of K n K n , the Cartesian product of... more We determine the minimum cardinality of an identifying code of K n K n , the Cartesian product of two cliques of same size. Moreover we show that this code is unique, up to row and column permutations, when n ≥ 5 is odd. If n ≥ 4 is even, we exhibit two distinct optimal identifying codes.
This paper presents a eletion game on graph, called the "Pic'arete". Two players al... more This paper presents a eletion game on graph, called the "Pic'arete". Two players alternately remove an edge of a given graph, one player getting one point each time the deletion of an edge isolates a vertex (so that you can get at most 2 points by edge deleted). When no edge remains, the player who has the maximum number of points is declared the winner. For instance, if we consider this game on the grid graph, we obtain a variant of the famous Dots and Boxes game. Our study consists in trying to make an estimate of the value of an initial game configuration, i.e., the difference between the maxima numbers of points attainable by both players. Finding a winning strategy will lead us to define a classification of game configurations, according to the parity of their number of edges, vertices, and sizes of hanging trees.
This paper presents a deletion game on graphs, called “Le Pic Arête”. Two players alternately rem... more This paper presents a deletion game on graphs, called “Le Pic Arête”. Two players alternately remove an edge of a given graph. A player gets one point each time he isolates a vertex (so that he gets at most two points by edge deleted). When no edge remains, the player with the maximum number of points is declared the winner. For instance, if we consider this game on a grid, we have a variant of the famous “Dots and Boxes” game. Our study consists of finding an approximation of the value of any game configuration, i.e., the difference between the numbers of points attainable by both players (when each player is supposed to play his best strategy).
Gravier and Maffray conjectured that every claw-free graph G satisfies $ch(G) = \chi(G) = \omega(... more Gravier and Maffray conjectured that every claw-free graph G satisfies $ch(G) = \chi(G) = \omega(G)$ (see [5] and [6]). We are interested in the restriction of this conjecture to the case where $G$ is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition it was proved in [7] that if $G$ is a claw-free perfect graph, with $\omega(G) \leq 3$ the conjecture is true. We prove it for the case $\omega(G) \leq 4$.
We introduce the concept of constant 2-labelling of a weighted graph and show how it can be used ... more We introduce the concept of constant 2-labelling of a weighted graph and show how it can be used to obtain periodic sphere packing. Roughly speaking, a constant 2-labelling of a weighted graph is a 2-coloring (black and white) of its vertex set which preserves the sum of the weight of black vertices under some automorphisms. In this manuscript, we study this problem on weighted complete graphs and on weighted cycles. Our results on cycles allow us to determine (r,a,b)-codes in Z^2 whenever |a-b|>4 and r>1.
The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study starte... more The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started a decade ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing colouring. A $d$-distinguishing colouring is a colouring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of this invariant. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb{N}^*$ colours. Alternatively, the two players choose a vertex of $G$ and colour it with one of the $d$ colours. The game ends when all the vertices have been coloured. Then the Gentle wins if the colouring is $d$-distinguishing and the Rascal wins otherwise. This game leads to a definition of two new invariants for a graph $G$, which are the minimum numbers of colours needed to ensure that the Gentle has a winning strategy...
We consider the problem of computing identifying codes of graphs and its fractional relaxation. T... more We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.
Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ ... more Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator. We give a construction of a family of circulant graphs of order $n$ and we show that this class has distinct distinguishing numbers and these lasters are not depending on $n$.
Ars Combinatoria -Waterloo then Winnipeg-
We consider the class of impartial combinatorial games for which the set of possible moves strict... more We consider the class of impartial combinatorial games for which the set of possible moves strictly decreases. Each game of this class can be considered as a domination game on a certain graph, called the move-graph. We analyze this equivalence for several families of combinatorial games and introduce an interesting graph operation, called twin and match, that preserves the Grundy value. We then study another game on graphs related to the dots and boxes game and propose a way to solve it.
Discrete Mathematics, 1995
In this communication the domination number of the cross product of an elementary path with the c... more In this communication the domination number of the cross product of an elementary path with the complement of another path is exactly determined and some inequalities for general cases are deduced. The paper ends with a Vizing-like conjecture relating the domination number of the cross product of G and G′ with the product of the corresponding ones.
Abstract: The -number of a graph G is the minimum value such that G admits alabeling with labels ... more Abstract: The -number of a graph G is the minimum value such that G admits alabeling with labels from f0; 1; : : : ; g where vertices at distance two get dierentlabels and adjacent vertices get labels that are at least two apart. Sierpinski graphsS(n; k) generalize the Tower of Hanoi graphs|the graph S(n; 3) is isomorphic tothe graph
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Papers by Sylvain Gravier