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Stéphane Bessy

Publications and source records attributed to Stéphane Bessy.

35 records · Page 2Linked to original sources
arXiv2018

Girth, minimum degree, independence, and broadcast independence

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $α_b(G)$ of $G$ is the largest weight $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. It is known that $α(G)\leq α_b(G)\leq 4α(G)$ for every connected graph $G$, where $α(G)$ is the independence number of $G$. If $G$ has girth $g$ and minimum degree $δ$, we show that $α_b(G)\leq 2α(G)$ provided that $g\geq 6$ and $δ\geq 3$ or that $g\geq 4$ and $δ\geq 5$. Furthermore, we show that, for every positive integer $k$, there is a connected graph $G$ of girth at least $k$ and minimum degree at least $k$ such that $α_b(G)\geq 2\left(1-\frac{1}{k}\right)α(G)$. Our results imply that lower bounds on the girth and the minimum degree of a connected graph $G$ can lower the fraction $\frac{α_b(G)}{α(G)}$ from $4$ below $2$, but not any further.

math.CO↗
arXiv2018

Algorithmic aspects of broadcast independence

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $α_b(G)$ of $G$ is the largest weight $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. We describe an efficient algorithm that determines the broadcast independence number of a given tree. Furthermore, we show NP-hardness of the broadcast independence number for planar graphs of maximum degree four, and hardness of approximation for general graphs. Our results solve problems posed by Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi (2006), Hedetniemi (2006), and Ahmane, Bouchemakh, Sopena (2018).

math.CO↗
arXiv2018

(Arc-disjoint) cycle packing in tournament: classical and parameterized complexity

Given a tournament $T$, the problem MaxCT consists of finding a maximum (arc-disjoint) cycle packing of $T$. In the same way, MaxTT corresponds to the specific case where the collection of cycles are triangles (i.e. directed 3-cycles). Although MaxCT can be seen as the LP dual of minimum feedback arc set in tournaments which have been widely studied, surprisingly no algorithmic results seem to exist concerning the former. In this paper, we prove the NP-hardness of both MaxCT and MaxTT. We also show that deciding if a tournament has a cycle packing and a feedback arc set with the same size is an NP-complete problem. In light of this, we show that MaxTT admits a vertex linear-kernel when parameterized with the size of the solution. Finally, we provide polynomial algorithms for MaxTT and MaxCT when the tournament is sparse, that is when it admits a FAS which is a matching.

cs.DM↗
arXiv2018

Dynamic monopolies for interval graphs with bounded thresholds

For a graph $G$ and an integer-valued threshold function $τ$ on its vertex set, a dynamic monopoly is a set of vertices of $G$ such that iteratively adding to it vertices $u$ of $G$ that have at least $τ(u)$ neighbors in it eventually yields the vertex set of $G$. We show that the problem of finding a dynamic monopoly of minimum order can be solved in polynomial time for interval graphs with bounded threshold functions, but is NP-hard for chordal graphs allowing unbounded threshold functions.

cs.DM↗
arXiv2018

Degree-constrained 2-partitions of graphs

A $(δ\geq k_1,δ\geq k_2)$-partition of a graph $G$ is a vertex-partition $(V_1,V_2)$ of $G$ satisfying that $δ(G[V_i])\geq k_i$ for $i=1,2$. We determine, for all positive integers $k_1,k_2$, the complexity of deciding whether a given graph has a $(δ\geq k_1,δ\geq k_2)$-partition. We also address the problem of finding a function $g(k_1,k_2)$ such that the $(δ\geq k_1,δ\geq k_2)$-partition problem is ${\cal NP}$-complete for the class of graphs of minimum degree less than $g(k_1,k_2)$ and polynomial for all graphs with minimum degree at least $g(k_1,k_2)$. We prove that $g(1,k)=k$ for $k\ge 3$, that $g(2,2)=3$ and that $g(2,3)$, if it exists, has value 4 or 5.

cs.DS↗
arXiv2017

Out-colourings of Digraphs

We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an {\bf out-colouring}. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is ${\cal NP}$-complete. We show that for every choice of positive integers $r,k$ there exists a $k$-strong bipartite tournament which needs at least $r$ colours in every out-colouring. Our main results are on tournaments and semicomplete digraphs. We prove that, except for the Paley tournament $P_7$, every strong semicomplete digraph of minimum out-degree at least 3 has a 2-out-colouring. Furthermore, we show that every semicomplete digraph on at least 7 vertices has a 2-out-colouring if and only if it has a {\bf balanced} such colouring, that is, the difference between the number of vertices that receive colour 1 and colour 2 is at most one. In the second half of the paper we consider the generalization of 2-out-colourings to vertex partitions $(V_1,V_2)$ of a digraph $D$ so that each of the three digraphs induced by respectively, the vertices of $V_1$, the vertices of $V_2$ and all arcs between $V_1$ and $V_2$ have minimum out-degree $k$ for a prescribed integer $k\geq 1$. Using probabilistic arguments we prove that there exists an absolute positive constant $c$ so that every semicomplete digraph of minimum out-degree at least $2k+c\sqrt{k}$ has such a partition. This is tight up to the value of $c$.

cs.DM↗
arXiv2017

On the Kőnig-Egerváry Theorem for $k$-Paths

The famous Kőnig-Egerváry theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we consider the set ${\cal G}_k$ of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order $k$ equals the minimum order of a set of vertices intersecting all paths of order $k$. For $k\in \{ 3,4\}$, we give complete structural descriptions of the graphs in ${\cal G}_k$. Furthermore, for odd $k$, we give a complete structural description of the graphs in ${\cal G}_k$ that contain no cycle of order less than $k$. For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order $k$ and minimum sets of vertices intersecting all paths of order $k$.

math.CO↗
arXiv2017

Out-degree reducing partitions of digraphs

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq i\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show that this problem is polynomial-time solvable when $p\geq 2k$ and ${\cal NP}$-complete otherwise. The result for $k=1$ and $p=2$ answers a question posed in \cite{bangTCS636}. We also determine, for all fixed non-negative integers $k_1,k_2,p$, the complexity of deciding whether a given digraph of maximum out-degree $p$ has a $2$-partition $(V_1,V_2)$ such that the digraph induced by $V_i$ has maximum out-degree at most $k_i$ for $i\in [2]$. It follows from this characterization that the problem of deciding whether a digraph has a 2-partition $(V_1,V_2)$ such that each vertex $v\in V_i$ has at least as many neighbours in the set $V_{3-i}$ as in $V_i$, for $i=1,2$ is ${\cal NP}$-complete. This solves a problem from \cite{kreutzerEJC24} on majority colourings.

cs.DM↗
arXiv2017

Bipartite spanning sub(di)graphs induced by 2-partitions

For a given $2$-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $B_G(V_1,V_2)$ of $G$ induced by those arcs/edges that have one end in each $V_i$. We determine, for all pairs of non-negative integers $k_1,k_2$, the complexity of deciding whether $G$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_i$ has at least $k_i$ (out-)neighbours in $V_{3-i}$. We prove that it is ${\cal NP}$-complete to decide whether a digraph $D$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_1$ has an out-neighbour in $V_2$ and each vertex in $V_2$ has an in-neighbour in $V_1$. The problem becomes polynomially solvable if we require $D$ to be strongly connected. We give a characterisation, based on the so-called strong component digraph of a non-strong digraph of the structure of ${\cal NP}$-complete instances in terms of their strong component digraph. When we want higher in-degree or out-degree to/from the other set the problem becomes ${\cal NP}$-complete even for strong digraphs. A further result is that it is ${\cal NP}$-complete to decide whether a given digraph $D$ has a $2$-partition $(V_1,V_2)$ such that $B_D(V_1,V_2)$ is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.

cs.DM↗
arXiv2017

Triangle packing in (sparse) tournaments: approximation and kernelization

Given a tournament T and a positive integer k, the C_3-Pakcing-T problem asks if there exists a least k (vertex-)disjoint directed 3-cycles in T. This is the dual problem in tournaments of the classical minimal feedback vertex set problem. Surprisingly C_3-Pakcing-T did not receive a lot of attention in the literature. We show that it does not admit a PTAS unless P=NP, even if we restrict the considered instances to sparse tournaments, that is tournaments with a feedback arc set (FAS) being a matching. Focusing on sparse tournaments we provide a (1+6/(c-1)) approximation algorithm for sparse tournaments having a linear representation where all the backward arcs have "length" at least c. Concerning kernelization, we show that C_3-Pakcing-T admits a kernel with O(m) vertices, where m is the size of a given feedback arc set. In particular, we derive a O(k) vertices kernel for C_3-Pakcing-T when restricted to sparse instances. On the negative size, we show that C_3-Pakcing-T does not admit a kernel of (total bit) size O(k^{2-ε}) unless NP is a subset of coNP / Poly. The existence of a kernel in O(k) vertices for C_3-Pakcing-T remains an open question.

cs.DS↗
arXiv2016

Bounds on the Burning Number

Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato, Janssen, and Roshanbin [Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22] define the burning number $b(G)$ of a graph $G$ as the smallest integer $k$ for which there are vertices $x_1,\ldots,x_k$ such that for every vertex $u$ of $G$, there is some $i\in \{ 1,\ldots,k\}$ with ${\rm dist}_G(u,x_i)\leq k-i$, and ${\rm dist}_G(x_i,x_j)\geq j-i$ for every $i,j\in \{ 1,\ldots,k\}$. For a connected graph $G$ of order $n$, they prove that $b(G)\leq 2\left\lceil\sqrt{n}\right\rceil-1$, and conjecture $b(G)\leq \left\lceil\sqrt{n}\right\rceil$. We show that $b(G)\leq \sqrt{\frac{32}{19}\cdot \frac{n}{1-ε}}+\sqrt{\frac{27}{19ε}}$ and $b(G)\leq \sqrt{\frac{12n}{7}}+3\approx 1.309 \sqrt{n}+3$ for every connected graph $G$ of order $n$ and every $0<ε<1$. For a tree $T$ of order $n$ with $n_2$ vertices of degree $2$, and $n_{\geq 3}$ vertices of degree at least $3$, we show $b(T)\leq \left\lceil\sqrt{(n+n_2)+\frac{1}{4}}+\frac{1}{2}\right\rceil$ and $b(T)\leq \left\lceil\sqrt{n}\right\rceil+n_{\geq 3}$. Furthermore, we characterize the binary trees of depth $r$ that have burning number $r+1$.

math.CO↗
arXiv2016

Extremal Values of the Chromatic Number for a Given Degree Sequence

For a degree sequence $d:d_1\geq \cdots \geq d_n$, we consider the smallest chromatic number $χ_{\min}(d)$ and the largest chromatic number $χ_{\max}(d)$ among all graphs with degree sequence $d$. We show that if $d_n\geq 1$, then $χ_{\min}(d)\leq \max\left\{ 3,d_1-\frac{n+1}{4d_1}+4\right\}$, and, if $\sqrt{n+\frac{1}{4}}-\frac{1}{2}>d_1\geq d_n\geq 1$, then $χ_{\max}(d)=\max\limits_{i\in [n]}\min\left\{ i,d_i+1\right\}$. For a given degree sequence $d$ with bounded entries, we show that $χ_{\min}(d)$, $χ_{\max}(d)$, and also the smallest independence number $α_{\min}(d)$ among all graphs with degree sequence $d$, can be determined in polynomial time.

math.CO↗
arXiv2015

Exponential Domination in Subcubic Graphs

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $γ_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$. In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If $G$ is a connected subcubic graph of order $n(G)$, then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq γ_e(G)\leq \frac{1}{3}(n(G)+2).$$ For every $ε>0$, there is some $g$ such that $γ_e(G)\leq εn(G)$ for every cubic graph $G$ of girth at least $g$. For every $0<α<\frac{2}{3\ln(2)}$, there are infinitely many cubic graphs $G$ with $γ_e(G)\leq \frac{3n(G)}{\ln(n(G))^α}$. If $T$ is a subcubic tree, then $γ_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic tree, $γ_e(T)$ can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs.

math.CO↗
arXiv2014

Two floor building needing eight colors

Motivated by frequency assignment in office blocks, we study the chromatic number of the adjacency graph of $3$-dimensional parallelepiped arrangements. In the case each parallelepiped is within one floor, a direct application of the Four-Colour Theorem yields that the adjacency graph has chromatic number at most $8$. We provide an example of such an arrangement needing exactly $8$ colours. We also discuss bounds on the chromatic number of the adjacency graph of general arrangements of $3$-dimensional parallelepipeds according to geometrical measures of the parallelepipeds (side length, total surface or volume).

math.CO↗
arXiv2011

Polynomial kernels for Proper Interval Completion and related problems

Given a graph G = (V,E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k pairs of (V \times V)\E such that the graph H = (V,E \cup F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research. First announced by Kaplan, Tarjan and Shamir in FOCS '94, this problem is known to be FPT, but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with at most O(k^5) vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem, admits a kernel with at most O(k^2) vertices, completing a previous result of Guo.

cs.DS↗
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↗