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Frédéric Havet

Publications and source records attributed to Frédéric Havet.

32 records · Page 2Linked to original sources
arXiv2019

Spanning eulerian subdigraphs in semicomplete digraphs

A digraph is eulerian if it is connected and every vertex has its in-degree equal to its out-degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs $(D,a)$ of a semicomplete digraph $D$ and an arc $a$ such that $D$ has a spanning eulerian subdigraph containing $a$. In particular, we show that if $D$ is $2$-arc-strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs $(D,a)$ of a semicomplete digraph $D$ and an arc $a$ such that $D$ has a spanning eulerian subdigraph avoiding $a$. In particular, we prove that every $2$-arc-strong semicomplete digraph has a spanning eulerian subdigraph avoiding any prescribed arc. We also prove the existence of a (minimum) function $f(k)$ such that every $f(k)$-arc-strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of $k$ arcs: we prove $f(k)\leq (k+1)^2/4 +1$, conjecture $f(k)=k+1$ and establish this conjecture for $k\leq 3$ and when the $k$ arcs that we delete form a forest of stars. A digraph $D$ is eulerian-connected if for any two distinct vertices $x,y$, the digraph $D$ has a spanning $(x,y)$-trail. We prove that every $2$-arc-strong semicomplete digraph is eulerian-connected. All our results may be seen as arc analogues of well-known results on hamiltonian cycles in semicomplete digraphs.

cs.DM↗
arXiv2018

On the unavoidability of oriented trees

A digraph is {\it $n$-unavoidable} if it is contained in every tournament of order $n$. We first prove that every arborescence of order $n$ with $k$ leaves is $(n+k-1)$-unavoidable. We then prove that every oriented tree of order $n$ ($n\geq 2$) with $k$ leaves is $(\frac{3}{2}n+\frac{3}{2}k -2)$-unavoidable and $(\frac{9}{2}n -\frac{5}{2}k -\frac{9}{2})$-unavoidable, and thus $(\frac{21}{8} n- \frac{47}{16})$-unavoidable. Finally, we prove that every oriented tree of order $n$ with $k$ leaves is $(n+ 144k^2 - 280k + 124)$-unavoidable.

cs.DM↗
arXiv2017

Finding a subdivision of a prescribed digraph of order 4

The problem of when a given digraph contains a subdivision of a fixed digraph $F$ is considered. Bang-Jensen et al. laid out foundations for approaching this problem from the algorithmic point of view. In this paper we give further support to several open conjectures and speculations about algorithmic complexity of finding $F$-subdivisions. In particular, up to 5 exceptions, we completely classify for which 4-vertex digraphs $F$, the $F$-subdivision problem is polynomial-time solvable and for which it is NP-complete. While all NP-hardness proofs are made by reduction from some version of the 2-linkage problem in digraphs, some of the polynomial-time solvable cases involve relatively complicated algorithms.

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

List Colouring Squares of Planar Graphs

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $Δ\ge8$ is at most $\bigl\lfloor\frac32Δ\bigr\rfloor+1$. We show that it is at most $\frac32 Δ(1+o(1))$ (where the $o(1)$ is as $Δ\to+\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs.

math.CO↗
arXiv2017

Complexity Dichotomies for the Minimum F-Overlay Problem

For a (possibly infinite) fixed family of graphs F, we say that a graph G overlays F on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of F as a spanning subgraph.While it is easy to see that the complete graph on |V(H)| overlays F on a hypergraph H whenever the problem admits a solution, the Minimum F-Overlay problem asks for such a graph with the minimum number of edges.This problem allows to generalize some natural problems which may arise in practice. For instance, if the family F contains all connected graphs, then Minimum F-Overlay corresponds to the Minimum Connectivity Inference problem (also known as Subset Interconnection Design problem) introduced for the low-resolution reconstruction of macro-molecular assembly in structural biology, or for the design of networks.Our main contribution is a strong dichotomy result regarding the polynomial vs. NP-hard status with respect to the considered family F. Roughly speaking, we show that the easy cases one can think of (e.g. when edgeless graphs of the right sizes are in F, or if F contains only cliques) are the only families giving rise to a polynomial problem: all others are NP-complete.We then investigate the parameterized complexity of the problem and give similar sufficient conditions on F that give rise to W[1]-hard, W[2]-hard or FPT problems when the parameter is the size of the solution.This yields an FPT/W[1]-hard dichotomy for a relaxed problem, where every hyperedge of H must contain some member of F as a (non necessarily spanning) subgraph.

cs.DS↗
arXiv2016

Colouring graphs with constraints on connectivity

A graph $G$ has maximal local edge-connectivity $k$ if the maximum number of edge-disjoint paths between every pair of distinct vertices $x$ and $y$ is at most $k$. We prove Brooks-type theorems for $k$-connected graphs with maximal local edge-connectivity $k$, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph $G$ with maximal local connectivity 3, outputs an optimal colouring for $G$. On the other hand, we prove, for $k \ge 3$, that $k$-colourability is NP-complete when restricted to minimally $k$-connected graphs, and 3-colourability is NP-complete when restricted to $(k-1)$-connected graphs with maximal local connectivity $k$. Finally, we consider a parameterization of $k$-colourability based on the number of vertices of degree at least $k+1$, and prove that, even when $k$ is part of the input, the corresponding parameterized problem is FPT.

math.CO↗
arXiv2016

Subdivisions of oriented cycles in digraphs with large chromatic number

An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).

math.CO↗
arXiv2016

$χ$-bounded families of oriented graphs

A famous conjecture of Gyárfás and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star $S$ and integer $k$, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size $k$ or it contains $S$ as an induced subgraph. As an evidence, we prove that for any oriented star $S$, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order $3$ or $S$ as an induced subdigraph. We then study for which sets ${\cal P}$ of orientations of $P_4$ (the path on four vertices) similar statements hold. We establish some positive and negative results.

cs.DM↗
arXiv2013

On wheel-free graphs

A wheel is a graph formed by a chordless cycle and a vertex that has at least three neighbors in the cycle. We prove that every 3-connected graph that does not contain a wheel as a subgraph is in fact minimally 3-connected. We give a new proof of a theorem of Thomassen and Toft: every graph that does not contain a wheel as a subgraph is 3-colorable.

math.CO↗
arXiv2013

Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs.

cs.DM↗
arXiv2010

k-L(2,1)-Labelling for Planar Graphs is NP-Complete for k >= 4

A mapping from the vertex set of a graph G = (V,E) into an interval of integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbour are mapped onto distinct integers. It is known that for any fixed k >= 4, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for k <= 3. For even k >= 8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k >= 4 by reduction from Planar Cubic Two-Colourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a proof of this.

math.CO↗