给定$\mathbb{R}^2$中的独立随机点$\mathcal{X}_n=\{X_1,...,X_n\}$，根据$\mathbb{R}^2$上的某个概率密度函数$f$画出$\mathcal{X}_n=\{X_1,...,X_n\}$，以及一个截断点$r_n>0$，我们构造了一个顶点集为$\mathcal{X}_n$的随机几何有向图$G(\mathcal{X}_n,\mathcal{Y}_n,r_n)$.每个顶点$X_i$在半径为$r_n$（以顶点$X_i$作为原点）的圆中被随机均匀地分配一个具有中心角$α$和倾角$Y_i$的扇区$S_i$。如果$X_j$落在$S_i$中，则存在从$X_i$到$X_j$的电弧。我们还引入了另一个随机几何有向图$G(\mathcal{X}_n,\mathcal{R}_n)$，它的顶点集为$\mathbb{R}^d$中的$\mathcal{X}_n=\{X_1,...,X_n\}$，$d\ge1$，如果$||X_i-X_j||<R_{n,i}$，则从$X_i$到$X_j$存在一条弧.此处$\{R_{n,i}\}_{i\ge1}$为i.i.d.。随机变量，我们可以取任意范数$||\cdot||$。本文研究了上述两种有向网络中的两类小子图--诱导图和孤立图，这有助于理解许多空间网络（如无线通信网络）的局部拓扑结构。本文给出了子图数的强大数定律，推广了Penrose [Random Geometric Graphs，Oxford University Press，2003]的结果.
英文摘要：
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 $G(\mathcal{X}_n,\mathcal{R}_n)$ with vertex set $\mathcal{X}_n=\{X_1,...,X_n\}$ in $\mathbb{R}^d$, $d\ge1$ and an arc present from $X_i$ to $X_j$ if $||X_i-X_j||<R_{n,i}$. Here $\{R_{n,i}\}_{i\ge1}$ are i.i.d. random variables and we may take an arbitrary norm $||\cdot||$. In this paper we investigate two kinds of small subgraphs--induced and isolated--in the above two directed networks, which contribute to understanding the local topology of many spatial networks, such as wireless communication networks. We give some strong laws of large numbers of subgraph counts thus extending those results of Penrose [Random Geometric Graphs, Oxford University Press, 2003].
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…
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…
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.
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 …
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…
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.
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.
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 …
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…
The digraph chromatic number of a directed graph $D$, denoted $χ_A(D)$, is the minimum positive integer $k$ such that there exists a partition of the vertices of $D$ into $k$ disjoint sets, each of which induces an acyclic subgraph. For any $m \geq 1$, a digraph is weakly $m$-degenerate if each of its induced subgraphs has a vertex of in-degree or out-degree less than $m$. We introduce a generalization of the digraph chromatic number, namely $χ_m(D)$, which is the minimum number of sets into which the vertices of a digraph $D$ can be partitioned so that each set induces a weakly $m$-degenerate…
