Section: New Results

Digraph theory

Participants : Julien Bensmail, Frédéric Havet, Nicolas Nisse, William Lochet.

We are putting an effort on understanding better directed graphs (also called digraphs) and partitioning problems, and in particular colouring problems. We also try to better the understand the many relations between orientations and colourings. We study various substructures and partitions in (di)graphs. For each of them, we aim at giving sufficient conditions that guarantee its existence and at determining the complexity of finding it.

Constrained ear decompositions in graphs and digraphs

Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously NP-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovász states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In [59], we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in $F$ is NP-complete for any finite set $F$ of positive integers. We also prove that, for any $k\ge 2$, deciding whether a graph admits an ear decomposition with all ears of length $0modk$ is NP-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in $F$ is NP-complete for any finite set $F$ of positive integers (and minimizing the number of handles of length in F is not approximable up to $n(1-\epsilon )$); for any $k\ge 2$, deciding whether a digraph admits a handle decomposition with all handles of length $0modk$ is NP-complete. Also, in contrast with the result of Lovász, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is NP-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is NP-complete, unless there exists $h\in 𝐍$ such that $A=\{1,\cdots ,h\}$.

Substructures in digraphs

We study substructures in digraphs. We study all kind of substructures: subdigraphs (induced or not), subdivision, immersion, minors, etc. We are both interested in the algorithmic point of view, that is determining the complexity of finding a (fixed or given) substructure in a given graph, and the structural point of view, that is finding sufficient conditions to guarantee the existence of a substructure.

In [32], we study the algorithmic complexity of the problem of deciding if a digraph contains a subdivision of a fixed digraph $F$. 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.

In [25], [22] we study conditions under which a digraph contain a subdivision of an oriented cycle. 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). We study the existence of more general structures than cycles. A $(k_1+k_2)$-bispindle is the union of $k_1$ $(x,y)$-dipaths and $k_2$ $(y,x)$-dipaths, all these dipaths being pairwise internally disjoint. The above-mentioned results on cycle with two blocks [25] can be restated as follows: for every $(1,1)$-bispindle $B$, there exists an integer $k$ such that every strongly connected digraph with chromatic number greater than $k$ contains a subdivision of $B$. In [21], we investigate generalizations of this result by first showing constructions of strongly connected digraphs with large chromatic number without any $(3,0)$-bispindle or $(2,2)$-bispindle. We then consider $(2,1)$-bispindles. Let $B(k_1,k_2;k_3)$ denote the $(2,1)$-bispindle formed by three internally disjoint dipaths between two vertices $x$, $y$, two $(x,y)$-dipaths, one of length $k_1$ and the other of length $k_2$, and one $(y,x)$-dipath of length $k_3$. We conjecture that for any positive integers $k_1$, $k_2$, $k_3$, there is an integer $g(k_1,k_2,k_3)$ such that every strongly connected digraph with chromatic number greater than $g(k_1,k_2,k_3)$ contains a subdivision of $B(k_1,k_2;k_3)$. As evidence, we prove this conjecture for $k_2=1$ (and $k_1$, $k_3$ arbitrary).

In [36], we prove the existence of a function $h\left(k\right)$ such that every simple digraph with minimum outdegree greater than $h\left(k\right)$ contains an immersion of the transitive tournament on k vertices. This solves a conjecture of Devos, McDonald, Mohar and Scheide  [72].

In [3], we study $\chi$-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 present some results and open problems about extensions of this conjecture to oriented graphs. In particular, 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 $𝒫$ of orientations of $P_4$ (the path on four vertices) similar statements hold. We establish some positive and negative results.

Partitions of digraphs

We also study partitions of digraphs. Again we are interested in the algorithmic point of view, that is determining the complexity of finding a partition satisfying some properties in a digraph, and the structural point of view, that is finding sufficient conditions to guarantee the existence of such a partition.

For a given 2-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study in [7] 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 $V_i$, $i\in \{1,2\}$. 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 $V_i$ (for $i\in \{1,2\}$) has at least $k_i$ (out-)neighbours in $V_{3-i}$. We prove that it is 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 characterization of the structure of $\mathrm{𝒩𝒫}$-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 NP-complete even for strong digraphs. A further result is that it is 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.

The dichromatic number $\overrightarrow{\chi }\left(D\right)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In [14], we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's Conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as $\Omega \left({log}_{2}n\right)$. We finally give a Brooks-type upper bound on the list dichromatic number of digon-free digraphs.