We generalize the class of split graphs to the directed case and show that these split digraphs can be identified from their degree sequences. The first degree sequence characterization is an extension of the concept of splittance to directed graphs, while the second characterization says a digraph is split if and only if its degree sequence satisfies one of the Fulkerson inequalities (which determine when an integer-pair sequence is digraphic) with equality.
In this paper we consider the directed path-width and directed tree-width of recursively defined digraphs. As an important combinatorial tool, we show how the directed path-width and the directed tree-width can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs. These results imply the equality of directed path-width and directed tree-width for all digraphs which can be defined by these four operations. This allows us to show a linear-time solution for computing the directed path-width and directed tree-width of all these digrap…
Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investigate directed modular width as a parameter, which is closer to the concept of clique-width. We investigate applications of modular decompositions of directed graphs to a wide range of algorithmic problems and derive FPT-algorithms for several well-kno…
Computing the directed path-width of a directed graph is an NP-hard problem. Even for digraphs of maximum semi-degree 3 the problem remains hard. We propose a decomposition of an input digraph G=(V,A) by a number k of sequences with entries from V, such that (u,v) in A if and only if in one of the sequences there is an occurrence of u appearing before an occurrence of v. We present several graph theoretical properties of these digraphs. Among these we give forbidden subdigraphs of digraphs which can be defined by k=1 sequence, which is a subclass of semicomplete digraphs. Given the decompositi…
A classical result by Erdos and Posa states that there is a function $f: {\mathbb N} \rightarrow {\mathbb N}$ such that for every $k$, every graph $G$ contains $k$ pairwise vertex disjoint cycles or a set $T$ of at most $f(k)$ vertices such that $G-T$ is acyclic. The generalisation of this result to directed graphs is known as Younger's conjecture and was proved by Reed, Robertson, Seymour and Thomas in 1996. This so-called Erdos-Posa-property can naturally be generalised to arbitrary graphs and digraphs. Robertson and Seymour proved that a graph $H$ has the Erdos-Posa-property if, and only if…
In the directed detour problem one is given a digraph $G$ and a pair of vertices $s$ and~$t$, and the task is to decide whether there is a directed simple path from $s$ to $t$ in $G$ whose length is larger than $\mathsf{dist}_{G}(s,t)$. The more general parameterized variant, directed long detour, asks for a simple $s$-to-$t$ path of length at least $\mathsf{dist}_{G}(s,t)+k$, for a given parameter $k$. Surprisingly, it is still unknown whether directed detour is polynomial-time solvable on general digraphs. However, for planar digraphs, Wu and Wang~[Networks, '15] proposed an $\mathcal{O}(n^3…
A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be its dichromatic number. This quantity has been widely studied in the last decades and can be considered as a natural directed analogue of the chromatic number of a graph. A digraph D is called even if for every 0-1-weighting of the edges it contains a directed cycle of even total weight. We show that every non-even digraph has dichromatic number at most 2 a…
By generalizing a recent result of Hong, Liu, and Lai on characterizing the degree-sequences of simple strongly connected directed graphs, a characterization is provided for degree-sequences of simple $k$-node-connected digraphs. More generally, we solve the directed node-connectivity augmentation problem when the augmented digraph is degree-specified and simple. As for edge-connectivity augmentation, we solve the special case when the edge-connectivity is to be increased by one and the augmenting digraph must be simple.
For integers m,k >= 1, we investigate the maximum size of a directed cut in directed graphs in which there are m edges and each vertex has either indegree at most k or outdegree at most k.
Mader proved that every strongly $k$-connected $n$-vertex digraph contains a strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, where the equality holds for the complete bipartite digraph ${DK}_{k,n-k}$. For dense strongly $k$-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly $k$-connected $n$-vertex digraph $D$ contains a strongly $k$-connected spanning subgraph with at most $kn + 800k(k+\overlineΔ(D))$ edges, where $\overlineΔ(D)$ denotes the maximum degree of the complement of the underlying undirected gra…
We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors. Moreover, we show that there is a linear relation between the hypertree-width of the dual of the cycle hypergraph of D, i. e. the hypergraph with vertices V (D) where every hyperedge corresponds to a directed cycle in D, and the directed treewidth of D. Based on this we show that a digraph has directed treewidth one if and only if its cycle hypergraph is a hypertree.
Let $D=(V,E)$ be a strongly connected digraph and let $u ,v\in V(D)$. The maximum distance $md (u,v)$ is defined as\\ $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length of a shortest directed $u-v$ path in $D$. This is a metric. The boundary, contour, eccentric and peripheral sets of a strong digraph $D$ with respect to this metric have been defined, and the above said metrically defined sets of a large strong digraph $D$ have been investigated in terms of the factors in its prime factor decomposition with respect to Cartesian …
Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order $n$ for which the size of the maximum acyclic set is at most $\lceil \frac{n+1}{2} \rceil$, for any $n$. This disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.
The three standard products (the Cartesian, the direct and the strong product) of undirected graphs have been wellinvestigated, unique prime factor decomposition (PFD) are known and polynomial time algorithms have been established for determining the prime factors. For directed graphs, unique PFD results with respect to the standard products are known. However, there is still a lack of algorithms, that computes the PFD of directed graphs with respect to the direct and the strong product in general. In this contribution, we focus on the algorithmic aspects for determining the PFD of directed gr…
In [Directed tree-width, J. Combin. Theory Ser. B 82 (2001), 138-154] we introduced the notion of tree-width of directed graphs and presented a conjecture, formulated during discussions with Noga Alon and Bruce Reed, stating that a digraph of huge tree-width has a large "cylindrical grid" minor. Here we prove the conjecture for planar digraphs, but many steps of the proof work in general. This is an unedited and unpolished manuscript from October 2001. Since many people asked for copies we are making it available in the hope that it may be useful. The conjecture was proved by Kawarabayashi and…
We investigate structural and algorithmic advantages of a directed version of the well-researched class of distance-hereditary graphs. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define directed twin-distance-hereditary graphs, which can be constructed by several twin and pendant vertex operations analogously to undirected distance-hereditary graphs and which still preserves the distance hereditary property. We give a characterization by forbidden induced subdigraphs and place the class in the hierarchy, comparing it to related classes. We …
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, trees, DAGs, cycles, and bipartite graphs. We also provide the idomatic number for special cases of some of these families of digraphs.
Let $D$ be a strongly connected digraph. The average distance $\barσ(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $ρ(D)$ and proximity $π(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $π(D)$ and $ρ(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $π(T)=ρ(T)$ if and only if …
This paper is devoted to the study of directed graphs with extremal properties relative to certain metric functionals. We characterize up to isomorphism critical digraphs with infinite values of diameter, quasi-diameter, radius and quasi-radius. Moreover, maximal digraphs with finite values of radius and quasi-diameter are studied.
In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on $n$ vertices, all preferred extensions can be enumerated in $O^*(3^{n/3})$ time and there are directed graphs with $Ω(3^{n/3})$ preferred extensions. We give faster enumeration algorithms for directed graphs with at most $0.8004\cdot n$ vertices occurring in $2$-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time $O(1.2321^n…
We introduce the notion of z-topological orderings for digraphs. We prove that given a digraph G on n vertices admitting a z-topological order- ing, together with such an ordering, one may count the number of subgraphs of G that at the same time satisfy a monadic second order formula ϕ and are the union of k directed paths, in time f (ϕ, k, z) * n^O(k*z) . Our result implies the polynomial time solvability of many natural counting problems on digraphs admitting z-topological orderings for constant values of z and k. Concerning the relationship between z-topological orderability and other digra…
The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the canonical tree decomposition theorem for digraphs. More precisely, we construct directed tree-decompositions of digraphs that distinguish all their tangles of order $k$, for any fixed integer $k$, in polynomial time. As an application of this canonical tree-decomposition theorem, we provide the following result for the directed disjoint paths problem: For every fixe…
The concept of Randic index has been extended recently for a digraph. We prove that $2R(G)\leq \mathcal{E}(G)\leq 2\sqrt{Δ(G)} R(G)$, where $G$ is a digraph, and $R(G)$ denotes the Randic index, $\mathcal{E}(G)$ denotes the Nikiforov energy and $Δ(G) $ denotes the maximum degree of $G$. In both inequalities we describe the graphs for which the equality holds.
The notions of bounded expansion and nowhere denseness have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion classes. More specifically, we show that the directed Steiner tree problem is fixed-parameter tractable on…
We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive integers; denote by $S(v)$ the sum of labels over all neighbors of each vertex $v$. The labeling $f$ is called \emph{topological additive numbering} if $S(u) < S(v)$ for each arc $(u,v)$ of the digraph. The problem asks to find the minimum number $k$ for which $D$ has a topological additive numbering with labels belonging to $\{ 1, \ldots, k \}$, denoted by $η_t…
Suppose that $D=(V,E)$ is a strongly connected digraph. Let $u,v\in V(D)$. The maximum distance $md (u,v)$ is defined as $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length of a shortest directed $u-v$ path in $D$. This is a metric. The boundary, contour, eccentric and peripheral sets of a strong digraph $D$ are defined with respect to this metric. The main aim of this paper is to identify the above said metrically defined sets of a large strong digraph $D$ in terms of its prime factor decomposition with respect to cartesian pro…
The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that - every strongly connected $n$-vertex digraph $D$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; - if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph UG($D$) is $O(k\log k)$. Moreover, if the d…
We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If…
We consider monotonicity problems for graph searching games. Variants of these games - defined by the type of moves allowed for the players - have been found to be closely connected to graph decompositions and associated width measures such as path- or tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowing the robber to reach positions that have been cleared before. The monotonicity problem for graph searching games has intensely been studied in the literature, but for two types of games the problem wa…
The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that \begin{itemize} \item every strongly connected digraph $D$ of order $n$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; \item if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its und…
A spanning subgraph $F$ of a graph $G$ is called {\em perfect} if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. Alex Scott (Graphs \& Combin., 2001) proved that every connected graph $G$ contains a perfect forest if and only if $G$ has an even number of vertices. We consider four generalizations to directed graphs of the concept of a perfect forest. While the problem of existence of the most straightforward one is NP-hard, for the three others this problem is polynomial-time solvable. Moreover, every digraph with only…
A minimum dominating set for a digraph (directed graph) is a smallest set of vertices such that each vertex either belongs to this set or has at least one parent vertex in this set. We solve this hard combinatorial optimization problem approximately by a local algorithm of generalized leaf removal and by a message-passing algorithm of belief propagation. These algorithms can construct near-optimal dominating sets or even exact minimum dominating sets for random digraphs and also for real-world digraph instances. We further develop a core percolation theory and a replica-symmetric spin glass th…
