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Stéphan Thomassé

Publications and source records attributed to Stéphan Thomassé.

84 records · Page 5Linked to original sources
arXiv2013

Symmetric Determinantal Representations in Characteristic 2

This paper studies Symmetric Determinantal Representations (SDR) in characteristic 2, that is the representation of a multivariate polynomial P by a symmetric matrix M such that P=det(M), and where each entry of M is either a constant or a variable. We first give some sufficient conditions for a polynomial to have an SDR. We then give a non-trivial necessary condition, which implies that some polynomials have no SDR, answering a question of Grenet et al. A large part of the paper is then devoted to the case of multilinear polynomials. We prove that the existence of an SDR for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings. We develop some algorithms to test the factorizability in these rings and use them to find SDRs when they exist. Altogether, this gives us polynomial-time algorithms to factorize the polynomials in the quotient rings and to build SDRs. We conclude by describing the case of Alternating Determinantal Representations in any characteristic.

cs.CC↗
arXiv2013

Complements of nearly perfect graphs

A class of graphs closed under taking induced subgraphs is $χ$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $χ(G) \leq f(ω(G))$. We consider the following question initially studied in [A. Gy{á}rf{á}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $χ$-bounded class $\cal C$, is the class $\bar{C}$ $χ$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $χ$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $χ$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $χ$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $χ$-bounded. For every $j$, we construct a class of graphs $χ$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $χ$-bounded.

math.CO↗
arXiv2012

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: - Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. - Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. - If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.

cs.DS↗
arXiv2012

A Linear Vertex Kernel for Maximum Internal Spanning Tree

We present a polynomial time algorithm that for any graph G and integer k >= 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G' on at most 3k vertices and an integer k' such that G has a spanning tree with at least k internal vertices if and only if G' has a spanning tree with at least k' internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel. Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that "a hypergraph H contains a hypertree if and only if H is partition connected."

cs.DS↗
arXiv2011

Cyclic Orderings and Cyclic Arboricity of Matroids

We prove a general result concerning cyclic orderings of the elements of a matroid. For each matroid $M$, weight function $ω:E(M)\rightarrow\mathbb{N}$, and positive integer $D$, the following are equivalent. (1) For all $A\subseteq E(M)$, we have $\sum_{a\in A}ω(a)\le D\cdot r(A)$. (2) There is a map $ϕ$ that assigns to each element $e$ of $E(M)$ a set $ϕ(e)$ of $ω(e)$ cyclically consecutive elements in the cycle $(1,2,...,D)$ so that each set $\{e|i\inϕ(e)\}$, for $i=1,...,D$, is independent. As a first corollary we obtain the following. For each matroid $M$ so that $|E(M)|$ and $r(M)$ are coprime, the following are equivalent. (1) For all non-empty $A\subseteq E(M)$, we have $|A|/r(A)\le|E(M)|/r(M)$. (2) There is a cyclic permutation of $E(M)$ in which all sets of $r(M)$ cyclically consecutive elements are bases of $M$. A second corollary is that the circular arboricity of a matroid is equal to its fractional arboricity. These results generalise classical results of Edmonds, Nash-Williams and Tutte on covering and packing matroids by bases and graphs by spanning trees.

math.CO↗
arXiv2011

The Domination Number of Grids

In this paper, we conclude the calculation of the domination number of all $n\times m$ grid graphs. Indeed, we prove Chang's conjecture saying that for every $16\le n\le m$, $γ(G_{n,m})=\lfloor\frac{(n+2)(m+2)}{5}\rfloor -4$.

cs.DM↗
arXiv2010

Multicut is FPT

Let $G=(V,E)$ be a graph on $n$ vertices and $R$ be a set of pairs of vertices in $V$ called \emph{requests}. A \emph{multicut} is a subset $F$ of $E$ such that every request $xy$ of $R$ is cut by $F$, ı.e. every $xy$-path of $G$ intersects $F$. We show that there exists an $O(f(k)n^c)$ algorithm which decides if there exists a multicut of size at most $k$. In other words, the \M{} problem parameterized by the solution size $k$ is Fixed-Parameter Tractable. The proof extends to vertex multicuts.

cs.DS↗
arXiv2010

Edge Growth in Graph Cubes

We show that for every connected graph $G$ of diameter $\ge 3$, the graph $G^3$ has average degree $\ge 7/4 δ(G)$. We also provide an example showing that this bound is best possible. This resolves a question of Hegarty \cite{PH}.

math.CO↗
arXiv2010

Coloring dense graphs via VC-dimension

The Vapnik-Červonenkis dimension is a complexity measure of set-systems, or hypergraphs. Its application to graphs is usually done by considering the sets of neighborhoods of the vertices (cf. Alon et al. (2006) and Chepoi, Estellon, and Vaxes (2007)), hence providing a set-system. But the graph structure is lost in the process. The aim of this paper is to introduce the notion of paired VC-dimension, a generalization of VC-dimension to set-systems endowed with a graph structure, hence a collection of pairs of subsets. The classical VC-theory is generally used in combinatorics to bound the transversality of a hypergraph in terms of its fractional transversality and its VC-dimension. Similarly, we bound the chromatic number in terms of fractional transversality and paired VC-dimension. This approach turns out to be very useful for a class of problems raised by Erdős and Simonovits (1973) asking for H-free graphs with minimum degree at least cn and arbitrarily high chromatic number, where H is a fixed graph and c a positive constant. We show how the usual VC-dimension gives a short proof of the fact that triangle-free graphs with minimum degree at least n/3 have bounded chromatic number, where $n$ is the number of vertices. Using paired VC-dimension, we prove that if the chromatic number of $H$-free graphs with minimum degree at least cn is unbounded for some positive c, then it is unbounded for all c<1/3. In other words, one can find H-free graphs with unbounded chromatic number and minimum degree arbitrarily close to n/3. These H-free graphs are derived from a construction of Hajnal. The large chromatic number follows from the Borsuk-Ulam Theorem.

math.CO↗
arXiv2009

Kernels for Feedback Arc Set In Tournaments

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.

cs.DS↗
arXiv2006

Convex cones and SAGBI bases of permutation invariants

Let G be a permutation group acting on {1,...,n}, and < be any admissible term order on the polynomial ring K[x_1,...,x_n]. We prove that the invariant ring K[x_1,...,x_n]^G of G has a finite SAGBI basis if, and only if, G is generated by reflections.

math.AC↗