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Let $D$ be a digraph, let $p \geq 1$ be an integer, and let $f: V(D) \to \mathbb{N}_0^p$ be a vector function with $f=(f_1,f_2,\ldots,f_p)$. We say that $D$ has an $f$-partition if there is a partition $(D_1,D_2,\ldots,D_p)$ into induced subdigraphs of $D$ such that for all $i \in [1,p]$, the digraph $D_i$ is weakly $f_i$-degenerate, that is, in every non-empty subdigraph $D'$ of $D_i$ there is a vertex $v$ such that $\min\{d_{D'}^+(v), d_{D'}^-(v)\} < f_i(v)$. In this paper, we prove that the condition $f_1(v) + f_2(v) + \ldots + f_p(v) \geq \max \{d_D^+(v),d_D^-(v)\}$ for all $v \in V(D)$ is almost sufficient for the existence of an $f$-partition and give a full characterization of the bad pairs $(D,f)$. Moreover, we describe a polynomial time algorithm that (under the previous conditions) either verifies that $(D,f)$ is a bad pair or finds an $f$-partition. Among other applications, this leads to a generalization of Brooks' Theorem as well as the list-version of Brooks' Theorem for digraphs, where a coloring of digraph is a partition of the digraph into acyclic induced subdigraphs. We furthermore obtain a result bounding the $s$-degenerate chromatic number of a digraph in terms of the maximum of maximum in-degree and maximum out-degree.

By a well known theorem of Robbins, a graph G has a strongly connected orientation if and only if G is 2-edge-connected and it is easy to find, in linear time, either a cut edge of G or a strong orientation of G. A result of Durand de Gevigny shows that for every k ≥ 3 it is NP-hard to decide if a given graph G has a k-strong orientation. Thomassen showed that one can check in polynomial time whether a given graph has a 2-strong orientation. This implies that for a given digraph D we can determine in polynomial time whether we can reorient (=reverse) some arcs of D = (V,A) to obtain a 2-strong digraph D' = (V,A'). This naturally leads to the question of determining the minimum number of such arcs to reverse before the resulting graph is 2-strong. In this paper we show that finding this number is NP-hard. If a 2-connected graph G has no 2-strong orientation, we may ask how many of its edges we may orient so that the resulting mixed graph is still 2-strong. Similarly, we may ask for a 2-edge-connected graph G how many of its edges we can orient such that the resulting mixed graph remains 2-arc-strong. We prove that when restricted to graphs satisfying suitable connectivity conditions, both of these problems are equivalent to finding the minimum number of edges we must double in a 2-edge-connected graph in order to obtain a 4-edge-connected graph. Using this, we show that all these three problems are NP-hard. Finally, we consider the operation of deorienting an arc uv of a digraph D meaning replacing it by an undirected edge between the same vertices. In terms of connectivity properties, this is equivalent to adding the opposite arc vu to D. We prove that for every ℓ ≥ 3 it is NP-hard to find the minimum number of arcs to deorient in a digraph D in order to obtain an ℓ-strong digraph D'.

A digraph is k‐strong if it has n ≥ k+1 vertices andeveryinduced subdigraph on at least n-k+1verticesis strongly connected. Atournament is a digraphwith no pair of nonadjacent vertices. We prove that everytournament on n ≥ k+1 vertices can be made k‐strong by adding no more than (k+1/2) arcs. This solves a conjecture from1994. A digraph is semicomplete if there is at least one arc between any pairof distinct vertices x, y. Sinceevery semicomplete digraph contains a spanning tournament, the result abovealso holds for semicomplete digraphs. Our result also implies that for every K≥2, every semicomplete digraph on atleast 3k−1vertices can be made k‐strong by reversing no more than (k+1/2) arcs.

A 2-edge-coloured graph G is supereulerian if G contains a spanning closed trail in which the edges alternate in colours. We show that for general 2-edge-coloured graphs it is NP-complete to decide whether the graph is supereulerian. An eulerian factor of a 2-edge-coloured graph is a collection of vertex-disjoint induced subgraphs which cover all the vertices of G such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test whether a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2-edge-coloured graph is (trail-)colour-connected if it contains a pair of alternating (u, v)-paths ((u, v)-trails) whose union is an alternating closed walk for every pair of distinct vertices u, v. We prove that a 2-edge-coloured complete bipartite graph is supereulerian if and only if it is colour-connected and has an eulerian factor. A 2-edge-coloured graph is M-closed if xz is an edge of G whenever some vertex u is joined to both x and z by edges of the same colour. M-closed 2-edge-coloured graphs, introduced in Contreras-Balbuena et al. (Discret Math Theoret Comput Sci 21:1, 2019), form a rich generalization of 2-edge-coloured complete graphs. We show that if G is obtained from an M-closed 2-edge-coloured graph H by substituting independent sets for the vertices of H, then G is supereulerian if and only if G is trail-colour-connected and has an eulerian factor. Finally we discuss 2-edge-coloured complete multipartite graphs and show that such a graph may not be supereulerian even if it is trail-colour-connected and has an eulerian factor.

In this paper, we exhibit connections between the following subjects: Tree packing in graphs and digraphs (both behave completely different), the rigidity matroid of a graph, Henneberg moves on trees, the conjectures of Thomassen and Matthews and Sumner, and (s,t)-orderings of digraphs. We do this by studying graphs which admit acyclic orientations that contain an out-branching and in-branching which are arc-disjoint (such an orientation is called good). A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. It is a well-known result due to Tutte and Nash-Williams, respectively, that every 4-edge-connected graph contains a spanning 2T-graph. Vertex-minimal 2T-graphs with at least two vertices which are known as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if (Formula presented.) is 2T-graph whose vertex set has a partition (Formula presented.) so that each (Formula presented.) induces a generic circuit (Formula presented.) of (Formula presented.) and the set of edges between different (Formula presented.) 's form a matching in (Formula presented.), then (Formula presented.) has a good orientation. We also obtain a characterization for the case when the set of edges between different (Formula presented.) 's form a double tree, that is, if we contract each (Formula presented.) to one vertex, and delete parallel edges we obtain a tree. All our proofs are constructive and imply polynomial algorithms for finding the desired good orderings and the pairs of arc-disjoint branchings which certify that the orderings are good. We identify a structure which can be used to certify that a given 2T-graph does not have a good orientation.

A digraph D is k-linked if it satisfies that for every choice of disjoint sets {x1,…,xk} and {y1,…,yk} of vertices of D there are vertex disjoint paths P1,…,Pk such that Pi is an (xi,yi)-path. Confirming a conjecture by Kühn et al., Pokrovskiy proved in 2015 that every 452k-strong tournament is k-linked and asked for a better linear bound. Very recently Meng et al. proved that every (40k−31)-strong tournament is k-linked. In this note we use an important lemma from their paper to give a short proof that every (13k−6)-strong tournament of minimum out-degree at least 28k−13 is k-linked.

Given a network modeled by a directed graph D = ( V , A ) , it is natural to ask whether we can partition the vertex set of D into two disjoint subsets V 1 , V 2 (called a 2-partition), such that the digraphs D [ V 1 ] , D [ V 2 ] induced by each of these has one of the two properties of interest. This question gives rise to a rich realm of combinatorial problems. The complexity of many such problems was determined in [2] , [3] . We analyze a subset of those problems from the viewpoint of parameterized complexity, and present a complete dichotomy of basic, natural properties. More precisely, given a directed graph D = ( V , A ) and two non-negative integers k 1 and k 2 , we seek a 2-partition ( V 1 , V 2 ) of the vertex set V such that | V 1 | ≥ k 1 , | V 2 | ≥ k 2 , and each of the subdigraphs induced by V 1 and V 2 has a structural property as defined by the problem at hand—for example, D [ V 1 ] is acyclic and D [ V 2 ] is strongly connected. Specifically, we consider the following eight structural properties: being strongly connected; being connected; having an out-branching; having an in-branching; having minimum degree at least one; having minimum semi-degree at least one; being acyclic; and being complete.

A directed graph D is semicomplete if for every pair x,y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D = (V,A) and a pair of natural numbers k and 𝓁, we are to decide whether there is a subset X of V of size k such that the largest strong connectivity component in D-X has at most 𝓁 vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for 𝓁 = 1. We study parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, 𝓁, 𝓁+k and n-𝓁. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^(16k)) but not in time O^*(2^o(k)) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^(16k)) implies the upper bound O^*(2^(16(n-𝓁))) for the parameter n-𝓁. We complement the latter by showing that there is no algorithm of time complexity O^*(2^o(n-𝓁)) unless ETH fails. Finally, we improve (in dependency on 𝓁) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter 𝓁+k on general digraphs from O^*(2^O(k𝓁 log (k𝓁))) to O^*(2^O(klog (k𝓁))). Note that Drange, Dregi and van 't Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^o(klog 𝓁)) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^o(klog k))., LIPIcs, Vol. 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), pages 2:1-2:16

A strong arc decomposition of a digraph D=(V,A) is a decomposition of its arc set A into two disjoint subsets A1 and A2 such that both of the spanning subdigraphs D1=(V,A1) and D2=(V,A2) are strong. Let T be a digraph with t vertices u1.....,ut and let H1....,Ht be digraphs such that Hi has vertices ui,j,. 1≤Ji≤ni Then the composition Q=T[H1...,Hi] is a digraph with vertex set (Formula presented.) and arc set (Formula presented.) We obtain a characterization of digraph compositions Q=T[H1...,Ht] which have a strong arc decomposition when T is a semicomplete digraph and each Hi is an arbitrary digraph. Our characterization generalizes a characterization by Bang-Jensen and Yeo (2003) of semicomplete digraphs with a strong arc decomposition and solves an open problem by Sun, Gutin, and Ai (2019) on strong arc decompositions of digraph compositions Q=T[H1..,Ht in which T is semicomplete and each (Formula presented.) is arbitrary. Our proofs are constructive and imply the existence of a polynomial algorithm for constructing a strong arc decomposition of a digraph Q=T[H1..,Ht, with T semicomplete, whenever such a decomposition exists.

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s 1 , s 2 , t 1 , t 2 whether D contains a pair of arc-disjoint paths P 1 , P 2 such that P i is an ( s i , t i ) -path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [8] . Recently it was shown [3] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair of constants k 1 , k 2 , there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P 1 , P 2 such that P i is an ( s i , t i ) -path of length no more than d ( s i , t i ) + k i , for i = 1 , 2 , where d ( s i , t i ) denotes the length of the shortest ( s i , t i ) -path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P 1 , P 2 to the weak 2-linkage problem where each path P i has length at most d ( s i , t i ) + c log 1 + ϵ ⁡ n for some constant c.

The dichromatic number $\overrightarrow{\chi}(D)$ of a digraph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class induces an acyclic subdigraph of $D$. A digraph $D$ is $k$-critical if $\overrightarrow{\chi}(D) = k$ but $\overrightarrow{\chi}(D') < k$ for all proper subdigraphs $D'$ of $D$. We examine methods for creating infinite families of critical digraphs, the Dirac join and the directed and bidirected Hajós join. We prove that a digraph $D$ has dichromatic number at least $k$ if and only if it contains a subdigraph that can be obtained from bidirected complete graphs on $k$ vertices by directed Hajós joins and identifying non-adjacent vertices. Building upon that, we show that a digraph $D$ has dichromatic number at least $k$ if and only if it can be constructed from bidirected $K_k$'s by using directed and bidirected Hajós joins and identifying non-adjacent vertices (so called Ore joins), thereby transferring a well-known result of Urquhart to digraphs. Finally, we prove a Gallai-type theorem that characterizes the structure of the low vertex subdigraph of a critical digraph, that is, the subdigraph, which is induced by the vertices that have in-degree $k-1$ and out-degree $k-1$ in $D$.

We give an FPT algorithm for deciding whether the vertex set of a digraph D can be partitioned into two disjoint sets V 1 , V 2 such that the digraph D [ V 1 ] induced by V 1 has a vertex that can reach all other vertices by directed paths, the digraph D [ V 2 ] has no vertex of in-degree zero and | V i | ≥ k i , where k 1 , k 2 are part of the input. This settles an open problem from [1] , [4] .

International audience; We prove that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B+$ and an in-branching $B−$ which are arc-disjoint (we call such branchings a good pair). This is best possible in terms of the arc-connectivity as there are infinitely many strong digraphs with independence number 2 and arbitrarily high minimum in- and out-degrees that have no good pair. The result settles a conjecture by Thomassen for digraphs of independence number 2. We prove that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and give an example of a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has no good pair. We also show that there are infinitely many digraphs with independence number 7 and arc-connectivity 2 that have no good pair. Finally we pose a number of open problems.

A 2-partition of a digraph D is a partition ( V 1 , V 2 ) of V ( D ) into two disjoint non-empty sets V 1 and V 2 such that V 1 ∪ V 2 = V ( D ) . A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following: • For all integers k 1 , k 2 ≥ 1 and k 1 + k 2 ≥ 3 it is NP-complete to decide whether a given digraph D has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 has out-degree at least k i for i = 1 , 2 . • For every fixed choice of integers α , k 1 , k 2 ≥ 1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 has out-degree at least k i for i = 1 , 2 . • For every fixed integer k ≥ 1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition ( V 1 , V 2 ) such that D 〈 V 1 〉 has out-degree at least one and D 〈 V 2 〉 has in-degree at least k. • It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 is a strong tournament.

We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is (Formula presented.) -complete. We show that for every choice of positive integers (Formula presented.) there exists a (Formula presented.) -strong bipartite tournament, which needs at least (Formula presented.) colours in every out-colouring. Our main results are on tournaments and semicomplete digraphs. We prove that, except for the Paley tournament (Formula presented.), every strong semicomplete digraph of minimum out-degree at least three has a 2-out-colouring. Furthermore, we show that every semicomplete digraph on at least seven vertices has a 2-out-colouring if and only if it has a 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 (Formula presented.) of a digraph (Formula presented.) so that each of the three digraphs induced by respectively, the vertices of (Formula presented.), the vertices of (Formula presented.) and all arcs between (Formula presented.) and (Formula presented.), have minimum out-degree (Formula presented.) for a prescribed integer (Formula presented.). Using probabilistic arguments, we prove that there exists an absolute positive constant (Formula presented.) so that every semicomplete digraph of minimum out-degree at least (Formula presented.) has such a partition. This is tight up to the value of (Formula presented.).

A $(\delta\geq k_1,\delta\geq k_2)$-partition of a graph $G$ is a vertex-partition $(V_1,V_2)$ of $G$ satisfying that $\delta(G[V_i])\geq k_i$ for $i=1,2$. We determine, for all positive integers $k_1,k_2$, the complexity of deciding whether a given graph has a $(\delta\geq k_1,\delta\geq k_2)$-partition. We also address the problem of finding a function $g(k_1,k_2)$ such that the $(\delta\geq k_1,\delta\geq k_2)$-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., Comment: 13 pages, 2 figures

International audience; An antidirected trail in a digraph is a trail (a walk with no arc repeated) in which the arcs alternate between forward and backward arcs. An antidirected path is an antidirected trail where no vertex is repeated. We show that it is NP-complete to decide whether two vertices x, y in a digraph are connected by an antidirected path, while one can decide in linear time whether they are connected by an antidirected trail. A digraph D is antistrong if it contains an antidirected (x, y)-trail starting and ending with a forward arc for every choice of x, y ∈ V (D) . We show that antistrong connectivity can be decided in linear time. We discuss relations between antistrong connectivity and other properties of a digraph and show that the arc-minimal antistrong spanning subgraphs of a digraph are the bases of a matroid on its arc-set. We show that one can determine in polynomial time the minimum number of new arcs whose addition to D makes the resulting digraph the arc-disjoint union of k antistrong digraphs. In particular, we determine the minimum number of new arcs which need to be added to a digraph to make it antistrong. We use results from matroid theory to characterize graphs which have an antistrong orientation and give a polynomial time algorithm for constructing such an orientation when it exists. This immediately gives analogous results for graphs which have a connected bipartite 2-detachment. Finally, we study arc-decompositions of antistrong digraphs and pose several problems and conjectures.

This paper studies the problem of proper-walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph i.e. a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with $k$ colours for every possible value of $k$., Comment: 25 pages, 9 figures

A digraph is {\bf 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. A digraph is {\bf semicomplete} if it has no pair of non-adjacent vertices. A {\bf tournament} is a semicomplete digraph without directed cycles of length 2. Fraise and Thomassen \cite{fraisseGC3} proved that every $(k+1)$-strong tournament has a hamiltonian cycle which avoids any prescribed set of $k$ arcs. In \cite{bangsupereuler} the authors demonstrated that a number of results concerning vertex-connectivity and hamiltonian cycles in tournaments and have analogues when we replace vertex connectivity by arc-connectivity and hamiltonian cycles by spanning eulerian subdigraphs. They showed the existence of a smallest function $f(k)$ such that every $f(k)$-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of $k$ arcs. They proved that $f(k)\leq \frac{(k+1)^2}{4}+1$ and also proved that $f(k)=k+1$ when $k=2,3$. Based on this they conjectured that $f(k)=k+1$ for all $k\geq 0$. In this paper we prove that $f(k)\leq (\lceil\frac{6k+1}{5}\rceil)$.

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.

A famous conjecture of Gy\'arf\'as 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., Comment: 27 pages, 4 figures

The dicycle transversal number of a digraph D is the minimum size of a dicycle transversal of D, that is a set of vertices of D, whose removal from D makes it acyclic. An arc a of a digraph D with at least one cycle is a transversal arc if a is in every directed cycle of D (making acyclic). In [3] and [4], we completely characterized the complexity of following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in its underlying undirected graph such that . It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where ). In the remaining case (allowing arbitrarily many transversal vertices) the problem is NP-complete. In this article, we classify the complexity of the arc-analog of this problem, where we ask for a dicycle B and a cycle C that are arc-disjoint, but not necessarily vertex-disjoint. We prove that the problem is polynomially solvable for strong digraphs and for digraphs with a constantly bounded number of transversal arcs (but possibly an unbounded number of transversal vertices). In the remaining case (allowing arbitrarily many transversal arcs) the problem is NP-complete.

We study out-branchings rooted at a given vertex s with the extra requirement that these satisfy certain balance conditions, such as balancing the number of arcs out of s, the number of vertices in each subtree of the branching etc. We obtain polynomial algorithms for several of the problems and study the borderline between polynomial and NP-complete special cases of the problems.

We prove that every digraph of circumference l has DAG-width at most l. This is best possible and solves a recent conjecture from S. Kintali (ArXiv:1401.2662v1 [math.CO], January 2014).1 As a consequence of this result we deduce that the k-linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in J. Bang-Jensen, F. Havet, and A. K. Maia (Theor Comput Sci 562 (2014), 283–303). We also prove that the weak k-linkage problem (where we ask for arc-disjoint paths) is polynomially solvable for every fixed k in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is still open. Finally, we prove that the minimum spanning strong subdigraph problem is NP-hard on digraphs of DAG-width at most 5.

A subgraph T of a digraph D is called an out-tree if T is an oriented tree with just one vertex s of in-degree zero. A spanning out-tree is called an out-branching. A vertex x of an out-branching B is called a leaf if the out-degree of x is zero. This is a survey on out-branchings with minimum and maximum number of leaves covering both structural and algorithmic results.

This edited volume offers a detailed account on the theory of directed graphs from the perspective of important classes of digraphs, with each chapter written by experts on the topic.Outlining fundamental discoveries and new results obtained over recent years, this book provides a comprehensive overview of the latest research in the field. It covers core new results on each of the classes discussed, including chapters on tournaments, planar digraphs, acyclic digraphs, Euler digraphs, graph products, directed width parameters, and algorithms. Detailed indices ease navigation while more than 120 open problems and conjectures ensure that readers are immersed in all aspects of the field.Classes of Directed Graphs provides a valuable reference for graduate students and researchers in computer science, mathematics and operations research. As digraphs are an important modelling tool in other areas of research, this book will also be a useful resource to researchers working in bioinformatics, chemoinformatics, sociology, physics, medicine, etc.

The class of tournaments is by far the most well-studied class of digraphs with many deep and important results. Since Moon’s pioneering book in 1968 [146], the study of tournaments and their properties has flourished and research on tournaments is still a very active area. Often this research deals with the superclass of semicomplete digraphs which are digraphs with no pair of non-adjacent vertices (that is, contrary to tournaments, we allow directed cycles of length 2). In this chapter we cover a very broad range of results on tournaments and semicomplete digraphs from classical to very recent ones. In order to stimulate further research, we not only list a number of open problems, but also give a number of proofs which illustrate the diversity of proof techniques that have been applied. These range from elementary to quite advanced.

In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph G and an integer ruυ for every pair of vertices u, υ ∊ V(G). The objective is to construct a subgraph H of minimum weight which contains ruυ edge-disjoint (or node-disjoint) u-υ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. Consequently, there is a long line of research into exact-polynomial time algorithms as well as approximation algorithms for various restrictions of this problem.\ud \ud An important restriction of this problem is one where the connectivity demands are the same for every pair of vertices. In this paper, we first consider the edge-connectivity version of this problem which we call λ-Edge Connected Subgraph (λ-ECS). In this problem, the input is a λ-edge connected (di)graph G and an integer k and the objective is to check whether G contains a spanning subgraph H that is also λ-edge connected and H excludes at least k edges of G. In other words, we are asked to compute a maximum subset of edges, of cardinality at least k, which may be safely deleted from G without affecting its connectivity. If we replace λ-edge connectivity with λ-vertex connectivity we get the λ-Vertex Connected Subgraph (λ-VCS) problem.\ud \ud We show that λ-ECS is fixed-parameter tractable (FPT) for both graphs and digraphs even if the (di)graph has nonnegative real weights on the edges and the objective is to exclude from H, some edges of G whose total weight exceeds a prescribed value. In particular, we design an algorithm for the weighted variant of the problem with running time 2O(k log k) |V(G)|O(1). We follow up on this result and obtain a polynomial compression for λ-ECS on unweighted graphs. As a direct consequence of our results, we obtain the first FPT algorithm for the parameterized version of the classical Minimum Equivalent Graph (MEG) problem. We also show that λ-Ves is FPT on digraphs; however the problem on undirected graphs remains open. Finally, we complement our algorithmic findings by showing that SNDP is W[1]-hard for both arc and vertex connectivity versions on digraphs. The core of our algorithms is composed of new combinatorial results on connectivity in digraphs and undirected graphs.\ud \ud \ud \ud \ud

In this chapter we will provide most of the terminology and notation used in this book. Various examples, figures and results should help the reader to better understand the notions introduced in the chapter. The results covered in this chapter constitute a collection of simple yet important facts on digraphs. Most of our terminology and notation are standard. Therefore, some readers may proceed to other chapters after a quick look through this chapter (unfamiliar terminology and notation can be clarified later by consulting the indices supplied at the end of this book).

We prove that every tournament T with no three disjoint cycles contains a set X of at most four vertices such that T-X is acyclic.

A digraph is supereulerian if it contains a spanning eulerian subdigraph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we analyze a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomasse and the first author: if the arc-connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasitransitive and verify the analogous statement for undirected graphs.

The multicut problem with respect to the fixed pattern (di)graph Q is the following: given a (di)graph G, an integer k and a 1-1 mapping @p of the vertices of Q to a subset X of V(G); determine whether there exists a set S of at most k edges (arcs) of G such that G-S has no (@p(q),@p(q^'))-path for every edge (arc) qq^' of Q. We first prove a dichotomy result for the complexity (they are all either polynomial or NP-complete) for multicut problems in (di)graphs in terms of the pattern Q. Then we consider a variant where the pattern is a mixed graph M; the input is a digraph D, an integer k and a 1-1 mapping @p from V(M) to some subset of V(D) and the goal is to decide whether there exists a set S of at most k arcs of D that meets every (@p(q),@p(q^'))-path in D for every arc qq^' of M and every (@p(p),@p(p^'))-path in the underlying undirected graph of D for every edge pp^' of M. Again we prove a dichotomy result for the complexity of such mixed multicut problems in terms of the pattern M. It turns out that the only polynomial cases are those where the pattern is either a polynomial graph pattern or a polynomial digraph pattern. As soon as M contains both edges and arcs, the M-mixed cut problem becomes NP-complete. Finally, we prove that the directed feedback arc set problem is NP-complete, even in the class of digraphs which have a feedback vertex set of size 2.

Deciding whether a digraph contains a pair of arc-disjoint in- and out-branchings rooted at a specified vertex is a well-known NP-complete problem as proved by Thomassen, see . This problem has been shown to be polynomial time solvable for semicomplete digraphs and for quasi-transitive digraphs . In this article, we study the problem for locally semicomplete digraphs. We characterize locally semicomplete digraphs that contain a pair of arc-disjoint in- and out-branchings rooted at a specified vertex. Our proofs are constructive and imply the existence of a polynomial time algorithm for finding the desired branchings when they exist. Our results generalizes those from for semicomplete digraphs and solves an open problem from .

We prove that the weak k-linkage problem is polynomial for every fixed k for totally i¾?-decomposable digraphs, under appropriate hypothesis on i¾?. We then apply this and recent results by Fradkin and Seymour on the weak k-linkage problem for digraphs of bounded independence number or bounded cut-width to get polynomial algorithms for some classes of digraphs like quasi-transitive digraphs, extended semicomplete digraphs, locally semicomplete digraphs all of which contain the class of semicomplete digraphs as a subclass and directed cographs.

An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x != s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: 1) Given a 2-regular digraph $D$, decide if it contains two arc-disjoint branchings B^+_u, B^-_v. 2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u such that D remains connected after removing the arcs of B^+_u. Both problems are NP-complete for general digraphs. We prove that the first problem remains NP-complete for 2-regular digraphs, whereas the second problem turns out to be polynomial when we do not prescribe the root in advance. We also prove that, for 2-regular digraphs, the latter problem is in fact equivalent to deciding if $D$ contains two arc-disjoint out-branchings. We generalize this result to k-regular digraphs where we want to find a number of pairwise arc-disjoint spanning trees and out-branchings such that there are k in total, again without prescribing any roots., Comment: 9 pages, 7 figures

We prove that every tournament with minimum out-degree at least inline image contains k disjoint 3-cycles. This provides additional support for the conjecture by Bermond and Thomassen that every digraph D of minimum out-degree inline image contains k vertex disjoint cycles. We also prove that for every inline image, when k is large enough, every tournament with minimum out-degree at least inline image contains k disjoint cycles. The linear factor 1.5 is best possible as shown by the regular tournaments.

A star in an undirected graph is a tree in which at most one vertex has degree larger than one. A star forest is a collection of vertex disjoint stars. An out-star (in-star) in a digraph D is a star in the underlying undirected graph of D such that all edges are directed out of (into) the center. The problem of partitioning the edges of the underlying graph of a digraph D into two star forests F 0 and F 1 is known to be NP-complete. On the other hand, with the additional requirement for F0 and F 1 to be forests of out-stars the problem becomes polynomial (via an easy reduction to 2-SAT). In this article we settle the complexity of problems lying in between these two problems. Namely, we study the complexity of the related problems where we require each F i to be a forest of stars in the underlying sense and require (in different problems) that in D, F i is either a forest of out-stars, in-stars, out- or in-stars or just stars in the underlying sense.

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., Comment: 17 pages, 4 figures

International audience; We continue the study, initiated in [3], of the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties and given minimum cardinality. Let EE be the following set of properties of digraphs: E=E={strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper we determine, for all choices of P1,P2P1,P2 from EE and all pairs of fixed positive integers k1,k2k1,k2, the complexity of deciding whether a digraph has a vertex partition into two digraphs D1,D2D1,D2 such that DiDi has property PiPi and |V(Di)|≥ki|V(Di)|≥ki, i=1,2i=1,2. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the analogous problems when P1∈HP1∈H and P2∈H∪EP2∈H∪E, where H=H={acyclic, complete, arc-less, oriented (no 2-cycle), semicomplete, symmetric, tournament} were completely characterized in [3].

International audience; 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.

We prove that a number of natural problems concerning the existence of arc-disjoint directed and ''undirected'' (spanning) subdigraphs in a digraph are NP-complete. Among these are the following of which the first settles an open problem due to Thomasse (see e.g. Bang-Jensen and Gutin (2009) [1, Problem 9.9.7] and Bang-Jensen and Kriesell (2009) [5,4]) and the second settles an open problem posed in Bang-Jensen and Kriesell (2009) [5]. *Given a directed graph D and a vertex s of D; does D contain an out-branching B"s^+ rooted at s such that the digraph remains connected (in the underlying sense) after removing all arcs of B"s^+? *Given a strongly connected directed graph D; does D contain a spanning strong subdigraph D^' such that the digraph remains connected (in the underlying sense) after removing all arcs of D^'?

We prove that every 3-strong semicomplete digraph on at least 5 vertices contains a spanning 2-strong tournament. Our proof is constructive and implies a polynomial algorithm for finding a spanning 2-strong tournament in a given 3-strong semicomplete digraph. We also show that there are infinitely many (2k−2)-strong semicomplete digraphs which contain no spanning k-strong tournament and conjecture that every(2k−1)-strong semicomplete digraph which is not the complete digraph K2k∗ on 2k vertices contains a spanning k-strong tournament.

This paper addresses the central spanning tree problem (CTP). The problem consists in finding a spanning tree that minimizes the so-called robust deviation, i.e. deviation from a maximally distant tree. The distance between two trees is measured by means of the symmetric difference of their edge sets. The central tree problem is known to be NP-hard. We attack the problem with a hybrid heuristic consisting of: (1) a greedy construction heuristic to get a good initial solution and (2) fast local search improvement. We illustrate computationally efficiency of the proposed approach.

We survey results and open problems concerning the existence of (arc-)disjoint subdigraphs with certain properties in a digraph. Examples are arc-disjoint spanning strong subdigraphs and disjoint paths with prescribed endvertices.

We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote the following two sets of natural properties of digraphs: H = {acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E = {strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper, we determine the complexity of deciding, for any fixed pair of positive integers k 1 , k 2 , whether a given digraph has a vertex partition into two digraphs D 1 , D 2 such that | V ( D i ) | ź k i and D i has property P i for i = 1 , 2 when P 1 ź H and P 2 ź H ź E . We also classify the complexity of the same problems when restricted to strongly connected digraphs.The complexity of the 2-partition problems where both P 1 and P 2 are in E is determined in the companion paper 2.

A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer $$\ell $$l, we can test whether D contains $$\ell $$l arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc-Disjoint Branchings and Non-Disconnecting Out-Branching. In Arc-Disjoint Branchings (Non-Disconnecting Out-Branching), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems. Non-Disconnecting Out-Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel. Arc-Disjoint Branchings is FPT on strong digraphs. The algorithm for Non-Disconnecting Out-Branching runs in time $$2^{\mathcal {O}(k)}n^{\mathcal {O}(1)}$$2O(k)nO(1) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.