Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [directed-graphs]

For questions about directed graphs. In a directed graph, each edge is an ordered pair of vertices; we think of it as pointing from one to the other. Use with the (graph-theory) tag.

Directed graphs, or digraphs, are used to represent asymmetric relationships between a collection of objects.

Formally, a directed graph is defined as consisting of:

  • a set of vertices;
  • set of ordered pairs of vertices, called arcs or directed edges.

In diagrams, vertices are represented by points, and each directed edge is represented by an arrow from its first vertex to its second.

Some special classes of directed graphs include:

  • Tournaments: directed graphs in which, for every pair of vertices, an edge exists in exactly one direction between them.
  • Directed acyclic graphs (DAGs), which contain no directed cycles.
432 questions
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Blocking directed paths on a DAG with a linear number of vertex defects.

Let $G=(V,E)$ be a directed acyclic graph. Define the set of all directed paths in $G$ by $\Gamma$. Given a subset $W\subseteq V$, let $\Gamma_W\subseteq \Gamma$ be the set of all paths $\gamma\in\Gamma$ supported on $V\backslash W$ (i.e all…
Yonathan Touati
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How many nodes in the smallest $k$-dense graph?

Let's call a directed graph $k$-dense if: Each node has exactly two children (outgoing neighbors); Each two nodes have at least three different children (besides themselves); Each three nodes have at least four different children (besides…
Erel Segal-Halevi
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A constrained topological sort?

Suppose that one has a directed, acyclic graph G, and each vertex $v$ contains a (positive) value $a_v$. Additionally, let $r$ be a constant. For my purposes, $r>1$, but this might not matter. Let $n$ be the number of vertices in G and let…
Aaron
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How do you correctly reason that this directed graph is acyclic?

How can you correctly reason that this directed graph is acyclic? I can only visually say that this graph is acyclic because there is not a single path in the graph where the starting vertex is equal to the ending vertex. But is this actually a…
kathelk
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A game on a graph

Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take turns, with Alice starting. On each of Alice's turns, she directs one undirected edge of $G$. - On each of Bob's turns, he selects a positive integer $m$ such that $1…
shadow10
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Generalized nontransitive dice

Let $X_1, \ldots, X_n$ be a collection of random variables. Consider the directed graph with vertex set $\{ 1, 2, \ldots, n \}$ where there is a directed edge $i \to j$ if $\mathbb{P}(X_i > X_j) > \frac{1}{2}$. Question 1: What directed graphs can…
Qiaochu Yuan
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Writing the Laplacian matrix of directed graphs as a product

The Laplacian matrix of an undirected graph can be written as $M^T M$ with $M$ being the incidence matrix of the graph. This makes the (otherwise tedious) proof of Kirchhoff's theorem into a beautiful application of the Cauchy-Binet formula (and…
Gadi A
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What is a directed acyclic graph (DAG)?

I am reading this link on Wikipedia; it states the following definition is given for a DAG. Definition: A DAG is a finite, directed graph with no directed cycles. Reading this definition believes me to think that the digraph below would be a DAG…
W. G.
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How many directed graphs of size n are there where each vertex is the tail of exactly one edge?

In a research problem in an unrelated area, me and a student found it necessary to count the number of directed graphs with every vertex having one outward-pointing edge, with no restrictions on the number of edges pointing inward. Not being a graph…
Brian Rushton
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Laplacian of a directed weighted graph

I know that for a simple undirected graph $\mathcal{G}(V,E) $ the Laplacian matrix $L$ is defined as: $$ L:=D-A$$ where $D$ is the degree diagonal matrix and $A$ is the adjacency matrix of $\mathcal{G}$. My question is: how can I define the…
Leonardo Massai
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A procedure for sampling paths in a directed acyclic graph

Consider a directed acyclic graph $\mathcal{G} = (\mathcal{N},\mathcal{E})$. Assume that the graph is very large (on the order of 10000 nodes and edges). Let there be a set of nodes termed starting nodes denoted by…
jonem
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Embedding a directed graph in its complement

Let $\Gamma$ be a directed graph on $n$ vertices with at most $2n - 4$ edges. Is $\Gamma$ embeddable in its complement? In other words, does there exist a bijection $\alpha$ of $V(\Gamma)$ such that $(x,y) \in E(\Gamma)$ implies $(\alpha(x),…
darko
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Tour of chess king

Consider lame chess king that can move only one cell left, down and diagonal upright. Consider square chess board. Question: Can such a king visit all cells of a board (each cell only once) and end up in the cell to the left of the original? I've…
Grigori
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Dominating sets in tournaments; is $2^{n+1}-2$ tight?

A tournement is a directed graph such that for every pair of distinct vertices $\{x,y\}$, there is either an edge from $x$ to $y$ or from $y$ to $x$, but not both. I will use "$x\to y$" to mean "there is an edge from $x$ to $y$." A dominating set of…
Mike Earnest
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Maximum number of edges in a directed graph on $n$ vertices without cycles

What is the maximum number of edges in a directed graph with $n$ vertices (which has no cycles). Logically it should be $n-1$, however I don't know how to prove this... Any help would be appreciated, Thanks!
Lola1984
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