We discuss the recently introduced concept of k-in-out graphs, and provide a construction for k-in-out graphs for any positive integer k. We derive a lower bound for the number of vertices of a k-in-out graph for any positive integer k, and demonstrate that our construction meets this bound in all cases. For even k, we also prove our construction is optimal with respect to the number of edges, and results in a planar graph. Among the possible uses of in-out graphs, they can convert the generalized traveling salesman problem to the asymmetric traveling salesman problem, avoiding the "big M" issue present in most other conversions. We give constraints satisfied by all in-out graphs to assist cutting-plane algorithms in solving instances of traveling salesman problem which contain in-out graphs.
We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every $0 < γ< 1/2$ we find a constant $c=c(γ)$ such that the following holds. Let $G=(V,E)$ be a (pseudo)random directed graph on $n$ vertices, and let $G'$ be a subgraph of $G$ with $(1/2+γ)|E|$ edges. Then $G'$ contains a directed cycle of length at least $(c-o(1))n$. Moreover, there is a subgraph $G''$ of $G$ with $(1/2+γ-o(1))|E|$ edges that does not contain a cycle of length at least $cn$.
Let $H$ be a directed acyclic graph other than a rooted star. It is known that there are constants $c(H)$ and $C(H)$ such that the following holds for the complete directed graph $D_n$. There are at most $C\log n$ directed acyclic subgraphs of $D_n$ that cover every $H$-copy of $D_n$, while fewer than $c\log n$ directed acyclic subgraphs of $D_n$ do not cover all $H$-copies. Here this dichotomy is considerably strengthened. Let ${\vec G}(n,p)$ denote the random directed graph. The {\em fractional arboricity} of $H$ is $a(H) = max \{\frac{|E(H')|}{|V(H')|-1}\}$, where the maximum is over all no…
We describe a general approach of determining the distribution of spanning subgraphs in the random graph $\G(n,p)$. In particular, we determine the distribution of spanning subgraphs of certain given degree sequences, which is a generalisation of the $d$-factors, of spanning triangle-free subgraphs, of (directed) Hamilton cycles and of spanning subgraphs that are isomorphic to a collection of vertex disjoint (directed) triangles.
An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.
A strongly connected graph is strongly biconnected if after ignoring the direction of its edges we have an undirected graph with no articulation points. A 3-vertex strongly biconnected graph is a strongly biconnected digraph that has the property that deleting any two vertices in this graph leaves a strongly binconnected subgraph. Jaberi [11] presented approximation algorithms for minimum cardinality 2-vertex strongly biconnected directed subgraph problem. We will focus in this paper on polynomial time algorithms which we have implemented for producing spanning subgraphs that are 3-vertex stro…
Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two specific kinds of digraph H: the three-edge path, with the first and last edges both directed towards the interior; and stars (with many edges directed out and many directed in). Aboulker et al subsequently conjectured that the answer is affirmative in both cases. We give affirmative answers to both questions.
The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of the corresponding class of graphs. In this paper, we analyze the game of cops and robbers on $\Vec{H}$-free digraphs. However, it is not the same as the case of undirected graph. So we give a new concept ($\Vec{H}^*$-free) to get a similar conclusion about the case of undirected graph.
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then. In this paper we consider two related questions concerning rainbow subgraphs of complete, edge-coloured graphs and digraphs. In the first part, we show that every properly edge-coloured complete directed graph contains a directed rainbow cycle of length $n-O(n^{4/5})$. This is motivated by an old problem of Hahn and improves a result of Gyarfas and Sarkozy.…
We study the directed maximum common edge subgraph problem (DMCES) for the class of directed graphs that are finite, weakly connected, oriented, and simple. We use DMCES to define a metric on partially ordered sets that can be represented as weakly connected directed acyclic graphs. While most existing metrics assume that the underlying sets of the partial order are identical, and only the relationships between elements can differ, the metric defined here allows the partially ordered sets to be different. The proof that there is a metric based on DMCES involves the extension of the concept of …
Directed graphs have long been used to gain understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Fruct's Theorem. We investigate four inverse semigroups defined over undirected graphs constructed from the notions of subgraph, vertex induced subgraph, rooted tree induced subgraph, and rooted path induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these four inverse semigroups. Finally, we prove a characterization …
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 consider vertex decompositions of (di)graphs which appear in Automata Theory, and establish some their properties. Then we apply them to the problem of forbidden subgraphs.
We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed graphs. As a particular case, we obtain a relation linking the Hamiltonian paths and cycles of a graph to its dominating connected sets.
Dense regions in networks are an indicator of interesting and unusual information. However, most existing methods only consider simple, undirected, unweighted networks. Complex networks in the real-world often have rich information though: edges are asymmetrical and nodes/edges have categorical and numerical attributes. Finding dense subgraphs in such networks in accordance with this rich information is an important problem with many applications. Furthermore, most existing algorithms ignore the higher-order relationships (i.e., motifs) among the nodes. Motifs are shown to be helpful for dense…
Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connecti…
We prove a conjecture by Aboulker, Charbit and Naserasr by showing that every oriented graph in which the out-neighborhood of every vertex induces a transitive tournament can be partitioned into two acyclic induced subdigraphs. We prove multiple extensions of this result to larger classes of digraphs defined by a finite list of forbidden induced subdigraphs. We thereby resolve several special cases of an extension of the famous Gyárfás-Sumner conjecture to directed graphs by Aboulker et al.
Given independent random points $\mathcal{X}_n=\{X_1,...,X_n\}$ in $\mathbb{R}^2$, drawn according to some probability density function $f$ on $\mathbb{R}^2$, and a cutoff $r_n>0$ we construct a random geometric digraph $G(\mathcal{X}_n,\mathcal{Y}_n,r_n)$ with vertex set $\mathcal{X}_n$. Each vertex $X_i$ is assigned uniformly at random a sector $S_i$, of central angle $α$ with inclination $Y_i$, in a circle of radius $r_n$ (with vertex $X_i$ as the origin). An arc is present from $X_i$ to $X_j$, if $X_j$ falls in $S_i$. We also introduce another random geometric digraph $G(\mathcal{X}_n,\mat…
Unavoidable subgraphs in digraphs with large out-degrees
Charles University, Prague, Czech Republic（布拉格查理大学）；Computer Science Institute (IÚUK, MFF), Charles University（查理大学数学与物理学院计算机科学研究所（IÚUK））；Department of Applied Mathematics (KAM, MFF), Charles University（查理大学数学与物理学院应用数学系（KAM））
We prove that for every $n$, there is a graph $G$ with $χ(G) \geq n$ and $ω(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $ω(H) \leq 2$ satisfies $χ(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.
Given a directed graph G=(V,E) an independent set A of the vertices V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed graph has a quasi-kernel. The plain generalization for infinite graphs fails, even for tournaments. We investigate the following conjecture here: for any digraph G=(V,E) there is a a partition (V_0,V_1) of the vertex set such that the induced subgraph G[V_0] has a quasi-kernel and the induced subgraph G[V_1] has a quasi-sink.
Let S and T be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence S in one part and T in the other; equivalently, binary matrices with row sums S and column sums T. In particular, we find precise formulae for the probabilities that a given bipartite graph is edge-disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out-degrees S and in-degrees T. In each case, the graphs o…
We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward $p$-stars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of directly constraining edge and other subgraph densities comes from Radin and Sadun. Such modeling circumvents a phenomenon first made precise by Chatterjee and Diaconis: that in ERGMs it is often impossible to independently constrain edge and other subgraph densities. In all our models, we find that large gr…
The Hajnal--Szemerédi theorem states that for any integer $r \ge 1$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $δ(G) \ge (1 - 1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the complete graph on $r$ vertices. We prove a very general analogue of this result for directed graphs: for any integer $r \ge 4$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1 - 1/r)n - 1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size $r$ eac…
We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL properties). %In our discussion, essential sets, introduced by Fult…
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…
A $k$-majority digraph is a directed graph created by combining $k$ individual rankings on the same ground set to form a consensus where edges point in the direction indicated by a strict majority of the rankings. The $k$-majority digraph is used to model voting scenarios, where the vertices correspond to options ranked by $k$ voters. When $k$ is odd, the resulting digraph is always a tournament, called $k$-majority tournament. Let $f_k(n)$ be the minimum, over all $k$-majority tournaments with $n$ vertices, of the maximum order of an induced transitive sub-tournament. Recently, Milans, Schrei…
In this paper, motivated by a problem of Scott and a conjecture of Lee, Loh and Sudakov we consider bisections of directed graphs. We prove that every directed graph with $m$ arcs and minimum semidegree at least $d$ admits a bisection in which at least $\left(\frac{d}{2(2d+1)}+o(1)\right)m$ arcs cross in each direction. This provides an optimal bound as well as a positive answer to a question of Hou and Wu in a stronger form.
In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at least $s$ on at most $nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}$ vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with $n$ vertices and average degree at least $n^{1-\frac{2}{s}+\varepsilon}$ contains a subgraph of average degree at least $s$ on $O_{\varepsilon,s}(1)$ v…
University of Edinburgh（爱丁堡大学）；University of Cyprus（塞浦路斯大学）；Pontificia Universidad Católica de Chile（智利天主教大学）
In graph-based applications, a common task is to pinpoint the most important or ``central'' vertex in a (directed or undirected) graph, or rank the vertices of a graph according to their importance. To this end, a plethora of so-called centrality measures have been proposed in the literature. Such measures assess which vertices in a graph are the most important ones by analyzing the structure of the underlying graph. A family of centrality measures that are suited for graph databases has been recently proposed by relying on the following simple principle: the importance of a vertex in a graph …
In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented hypercube $\overrightarrow{Q_n}$ such that the induced subgraph $\overrightarrow{Q_n}[U]$ does not contain any copy of $\overrightarrow{F}$. We obtain the exact value of $ex_v(\overrightarrow{P_k}, \overrightarrow{Q_n})$ for the directed path $\overrightarrow{P_k}$, the exact value of $ex_v(\overrightarrow{V_2}, \overrightarrow{Q_n})$ for the…
Let $\F_p = \Z/p\Z$. The \emph{height} of a point $\mathbf{a}=(a_1,..., a_d) \in \F_p^d$ is $h_p(\mathbf{a}) = \min \left\{\sum_{i=1}^d (ka_i \mod p) : k=1,...,p-1\right\}.$ Explicit formulas and estimates are obtained for the values of the height function in the case $d=2,$ and these results are applied to the problem of determining the minimum number of edges the must be deleted from a finite directed graph so that the resulting subgraph is acyclic.
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…
Extracting structured subgraphs inside large graphs - often known as the planted subgraph problem - is a fundamental question that arises in a range of application domains. This problem is NP-hard in general, and as a result, significant efforts have been directed towards the development of tractable procedures that succeed on specific families of problem instances. We propose a new computationally efficient convex relaxation for solving the planted subgraph problem; our approach is based on tractable semidefinite descriptions of majorization inequalities on the spectrum of a symmetric matrix.…
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.
We study optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. We prove for $r$ fixed and $n$ large the condition to be $δ(G) \ge \frac{n+r-2}{2}$ when $nr \equiv 0 \pmod 2$. This answers a question of M.~Kriesell.
A group $G$ of permutations of a set $Ω$ is {\em primitive} if it acts transitively on $Ω$, and the only $G$-invariant equivalence relations on $Ω$ are the trivial and universal relations. A graph $Γ$ is {\em primitive} if its automorphism group acts primitively on its vertex set. A graph $Γ$ has {\em connectivity one} if it is connected and there exists a vertex $α$ of $Γ$, such that the induced graph $Γ\setminus \{α\}$ is not connected. If $Γ$ has connectivity one, a {\em block} of $Γ$ is a connected subgraph that is maximal subject to the condition that it does not have connectivity one. Th…
Boltzmann sampling and optimal exact-size sampling for directed acyclic graphs
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Université Caen Normandie, ENSICAEN, CNRS, Normandie Univ, GREYC UMR 6072（卡昂诺曼底大学，ENSICAEN，CNRS，诺曼底大学，GREYC联合研究实验室）
