References of "Vandomme, Elise"
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See detailIdentifying codes in vertex-transitive graphs
Gravier, Sylvain; Parreau, Aline ULg; Rottey, Sara et al

Conference (2014, July)

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the optimal integer and fractional solutions is between 1 and 2 log(|V|) where V is the ... [more ▼]

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the optimal integer and fractional solutions is between 1 and 2 log(|V|) 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. [less ▲]

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See detailA conjecture on the 2-abelian complexity of the Thue-Morse word
Rigo, Michel ULg; Parreau, Aline ULg; Vandomme, Elise ULg

Scientific conference (2014, January 20)

The Thue-Morse word is a well-known and extensively studied 2-automatic sequence. For example, it is trivially abelian periodic and its abelian complexity takes only two values. For an integer k, the k ... [more ▼]

The Thue-Morse word is a well-known and extensively studied 2-automatic sequence. For example, it is trivially abelian periodic and its abelian complexity takes only two values. For an integer k, the k-abelian complexity is a generalization of the abelian complexity, corresponding to the case where k=1. Formally, two words u and v of the same length are k-abelian equivalent if they have the same prefix (resp. suffix) of length k-1 and if, for all words x of length k, the numbers of occurrences of x in u and v are the same. This notion has received some recent interest, see the works of Karhumäki et al. The k-abelian complexity of an infinite word x maps an integer n to the number of k-abelian classes partitioning the set of factors of length n occurring in x. The aim of this talk is to study the 2-abelian complexity a(n) of the Thue-Morse word. We conjecture that a(n) is 2-regular in the sense of Allouche and Shallit. This question can be related to a work of Madill and Rampersad (2012) where the (1)-abelian complexity of the paper folding word is shown to be 2-regular. We will present some arguments supporting our conjecture. They are based on functions counting some subword of length 2 occuring in prefixes of the Thue-Morse word. [less ▲]

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See detailLinear formulation of identifying cdes in graphs
Vandomme, Elise ULg; Gravier, Sylvain; Parreau, Aline ULg

Poster (2013, September 09)

Identifying codes were introduced by Karpovsky, Chakrabarty and Levitin in 1998 and can be applied to locate fire in a building using sensors. Buildings are modelled by graphs with rooms as vertices. The ... [more ▼]

Identifying codes were introduced by Karpovsky, Chakrabarty and Levitin in 1998 and can be applied to locate fire in a building using sensors. Buildings are modelled by graphs with rooms as vertices. The placement of sensors in the rooms corresponds to choosing a subset of vertices. Finding a sensor-placement such that the location of a fire in one room can be precisely determined is equivalent to constructing an identifying code in the graph. These are dominating sets of vertices for which the closed neighbourhood of each vertex (i.e., the vertex and its neighbours) has a unique intersection with the set. The problem of finding an identifying code has been widely studied. Yet its formulation as an integer linear problem hasn't been much considered. Let G be a graph with vertex set V, to an identifying code $C\subseteq V$ of $G$ correspond weights x_u (x_u is 1 if u belongs to C, otherwise x_u is 0) satisfying the following : for all vertices u,v * the sum of the x_w for w in the closed neighbourhood of u is at least 1 * the sum of the x_w for w in the symmetric difference of the closed neighbourhoods of u and v is at least 1. Of course, it is interesting to find an identifying code with the smallest possible cardinality. But in general this is a NP-hard problem. A way to obtain bounds on the minimal cardinality is to consider the associated linear problem where the weights x_u are fractional. In the case of vertex-transitive graphs, the minimal cardinality for the fractional case can only take two values which depend on the number of vertices, the degree of the graph and the smallest symmetric difference between any two closed neighbourhoods. We show that for an infinite family of graphs the bound is tight and for another another the bound is far too be reached. [less ▲]

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See detailSome properties of abelian return words
Rigo, Michel ULg; Salimov, Pavel ULg; Vandomme, Elise ULg

in Journal of Integer Sequences (2013), 16

We investigate some properties of abelian return words as recently introduced by S. Puzynina and L. Q. Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in ... [more ▼]

We investigate some properties of abelian return words as recently introduced by S. Puzynina and L. Q. Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue--Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue--Morse word, the set of abelian derived sequences is infinite. [less ▲]

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See detail2-abelian complexity of the Thue-Morse sequence
Rigo, Michel ULg; Vandomme, Elise ULg

Scientific conference (2012, December 14)

Let k be an integer. Two words u and v of the same length are k-abelian equivalent, if they have the same prefix (resp. suffix) of length k-1 and if, for all words x of length k, the numbers of ... [more ▼]

Let k be an integer. Two words u and v of the same length are k-abelian equivalent, if they have the same prefix (resp. suffix) of length k-1 and if, for all words x of length k, the numbers of occurrences of x in u and v are the same. This notion has received some recent interest, see the works of Karhumäki et al. The k-abelian complexity of an infinite word x maps an integer n to the number of k-abelian classes partitioning the set of factors of length n occurring in x. The Thue-Morse word is a well-known and extensively studied 2-automatic sequence. It is trivially abelian periodic and its (1)-abelian complexity takes only two values. The aim of this talk is to explain how to compute the 2-abelian complexity a(n) of the Thue-Morse word, showing in particular that it is unbounded. We conjecture that a(n) is 2-regular in the sense of Allouche and Shallit. This question can be related to a recent work of Madill and Rampersad where the (1)-abelian complexity of the paper folding word is shown to be 2-regular. We will also explain why the usual logical framework introduced by Büchi and popularized by Bruyère (indeed, automatic sequences can be characterized in an extension of the Presburger arithmetic) and the recent work of Charlier et al. about "automatic theorem-proving" of properties of automatic sequences cannot be applied to these questions related to abelian complexity. [less ▲]

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See detailConstant 2-labelling of weighted cycles
Gravier, Sylvain; Vandomme, Elise ULg

Conference (2012, November)

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 ... [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. [less ▲]

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See detailSome properties of abelian return words (long abstract)
Rigo, Michel ULg; Salimov, Pavel ULg; Vandomme, Elise ULg

Conference (2012, September 11)

We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the ... [more ▼]

We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue–Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue–Morse word, the set of abelian derived sequences is infinite. [less ▲]

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See detailConstant 2-labelling of a graph
Gravier, Sylvain; Vandomme, Elise ULg

Conference (2012, September)

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 ... [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. [less ▲]

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See detailColoring of the infinite grid
Vandomme, Elise ULg; Gravier, Sylvain

Poster (2012, May 23)

We present a method that links (r,a,b)-codes in the infinite grid and particular colorings of weighted cycles, called constant 2-labellings.

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See detailSyntactic complexity of ultimately periodic sets of integers and application to a decision procedure
Lacroix, Anne ULg; Rampersad, Narad; Rigo, Michel ULg et al

in Fundamenta Informaticae (2012), 116

We compute the cardinality of the syntactic monoid of the language 0* rep_b(mN) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any ... [more ▼]

We compute the cardinality of the syntactic monoid of the language 0* rep_b(mN) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integers written in base b. We apply our results to a well studied problem: decide whether or not a b-recognizable set of integers is ultimately periodic. [less ▲]

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See detailSyntactic complexity of ultimately periodic sets of integers
Rigo, Michel ULg; Vandomme, Elise ULg

Conference (2011, June)

We compute the cardinality of the syntactic monoid of the language 0^∗rep_b(mN) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any ... [more ▼]

We compute the cardinality of the syntactic monoid of the language 0^∗rep_b(mN) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integers written in base b. We apply our results to some well studied problem: decide whether or not a b-recognizable sets of integers is ultimately periodic. [less ▲]

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See detailSyntactic complexity of ultimately periodic sets of integers
Rigo, Michel ULg; Vandomme, Elise ULg

in Lecture Notes in Computer Science (2011), 6638

We compute the cardinality of the syntactic monoid of the language 0^∗rep_b(mN) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any ... [more ▼]

We compute the cardinality of the syntactic monoid of the language 0^∗rep_b(mN) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integers written in base b. We apply our results to some well studied problem: decide whether or not a b-recognizable sets of integers is ultimately periodic. [less ▲]

Detailed reference viewed: 48 (22 ULg)