Questions tagged [directed-graphs]

A directed graph is a graph with directed edges. Loops and 2-cycles are usually allowed. See also the tag *quiver*.

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113 views

Directed version of this lemma

On a paper by Shoham Letzter, available Here, there's a lemma that says as follows: Lemma 0: For every graph $G$, there exist two disjoint sets $U,W\subseteq V(G)$ of equal size, such that there are ...
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Minimum of sums over degree products in a directed acyclic graph

My problem is the following: we have a graph $ G=(V,E)$. Having a total ordering $ \eta $ of the nodes, we give a direction to the edges such that $ (u,v) $ is directed from $u$ to $v$ iif $ \eta(u) &...
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128 views

Dominating sets in subtournaments of the Paley tournament

For a tournament $T$, let $\mathrm{dom}(T)$ be the order of a smallest dominating set in $T$. Let $q$ be a prime power congruent to 3 mod 4 and let $T_q$ be the Paley tournament on $q$ vertices. Is ...
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Does the random graph interpret the random directed graph?

The random graph is the Fraisse limit of the class of finite graphs, the random directed graph is the Fraisse limit of the class of directed graphs, a directed graph is just a set with a binary ...
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184 views

Convergence on iterating a piecewise function

Given the four functions $P_1$, $P_2$, $N_1$ and $N_2$ (which together is a piecewise function) each with domain and range as shown above: Is there an explanation as to why starting at any integer (...
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59 views

Name for a directed acyclic graph with no skip-level edges?

I'm looking at a specific class of DAGs, namely those DAGs such that any path from $u$ to $v$ has the same length. Informally, we don't allow "skip-level" edges. I understand these graphs ...
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279 views

Proving that every strongly connected tournament T on at least 4 vertices contains distinct vertices u, v such that T-u and T-v are strongly connected

I have a two part question: Is there a simple proof that every strongly connected tournament $T$ on $n\geq 4$ vertices contains distinct $u,v\in V(T)$ such that $T-u$ and $T-v$ are strongly connected?...
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87 views

Infinite recursive graphs and different ways to build them

I asked this question one week ago on MSE and has received no answer. Infinite directed graphs (graphs with countably many nodes and edges) have a number of different applications. They can be ...
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1answer
87 views

Using a poset or directed graph as input for a neural network [closed]

I'm not sure if this is the right community to post this in but I would appreciate any help. As the title states, I'm trying to train a neural network using some unconventional input. I'm wondering if ...
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Kernel perfect orientations of complete graphs

How can we create a kernel perfect orientation of a complete graph? A kernel of a graph is a set of vertices in a graph $G$, which absorbs other vertices, that is, has all the vertices in its ...
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2answers
170 views

Digraphs with exactly one Eulerian tour

I’ve been thinking about the following problem from Richard Stanley’s list of bijective proof problems (2009). There, this problem is said to lack a combinatorial solution. The problem is the ...
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1answer
123 views

Digraphs with unique walk of length $k$ between any two vertices

Let $G$ be a digraph such that there is an unique directed walk of length $k$ between any two vertices. Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $...
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125 views

Directed graph minor theorems

In proving the graph minor theorem, Robertson and Seymour proved a stronger statement, namely that the directed graph minor theorem is true, using the definition A directed graph is a minor of ...
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1answer
59 views

Has it already been shown that a digraph is $k$-connected if and only if there is a set of $k$ disjoint paths between every pair of vertices? [closed]

My question concerns graph theory. There is a conjecture I know to be true, but I am not sure whether it has been proven before. It is a fairly simple result. It may be well-known; I am not sure. ...
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72 views

Expected number of directed cycle in a directed complete graph

Consider the randomized, directed complete graph G = (V, E) where for each pair of vertices u, v ∈ V, we add either the directed edge (u → v) or the directed edge (v → u) chosen uniformly at random. ...
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1answer
66 views

Distance pairs in labeled directed graph

Suppose we have a simple directed graph with $n$ nodes and $m$ edges, and we label each edge from $1$ to $m$ (with distinct labels). Define the weighted "length" of a directed path to be the maximum ...
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On the number of connected functional digraphs recoverable from the preimage set size structure

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, $P_j=\left[f^{-j}(...
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104 views

Terminology for transforming a directed acyclic graph into a tree

I am looking for the term of converting a directed acyclic graph (DAG) into a tree by traversing its topologically ordered nodes and copying the subtrees of the nodes with in-degree $> 1$. Such a ...
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70 views

Discrete version of Helmholtz decomposition

In The curl of graphs and networks (Gustafson and Haray, 1984) it is claimed to be shown that any digraph $G$ can be decomposed as the sum of three graphs $U_1 + U_2 + U_3$, where $U_1$ is divergence-...
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1answer
56 views

Infinite directed paths in tournaments on $\omega$

Does every tournament on $\omega$ contain an infinite directed path that doesn't visit any vertex twice?
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1answer
136 views

Explicit upper bound on the number of simple rooted directed graphs on 𝑛 vertices?

Harary mentioned this problem in "The number of linear, directed, rooted, and connected graphs" on p. 455, l. 3–5, but a short and crisp upper bound is missing. I believe that someone must have ...
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236 views

When does a graph have a circular orientation? Or equivalently can anyone help me characterize this particular class of $3$-colorable perfect graphs?

Call an oriented digraph $D=(V,A)$ circular when for all $\small x,y,z\in V$ if $(x,y)\in A$ and $(y,z)\in A$ then $(z,x)\in A$ or equivalently if $D$ is any oriented digraph whose arc set is a ...
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215 views

Population of P people, where each person knows K others, how many people mutually know each other

If you have a population of $P$ people, where each person knows $K$ others within the population (does not have to be mutual, i.e., if I know you, you don't necessarily know me), and $1<K<P$, ...
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1answer
183 views

Characterizing relations by forbidden induced subsets

Working with relations in a purely set theoretic manner i.e. as just sets of ordered pairs, we see for any relation $R$ there exists unique inclusion minimal sets $A$ and $B$ such that $R\subseteq A\...
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391 views

Finding the max-value set of cycles in a weighted digraph

I am looking for the most efficient algorithm that can solve this problem: Given a directed graph with real-valued edge weights, find a set of directed cycles (no two cycles can share a vertex) that ...
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401 views

When does a graph have a minimally strong orientation?

Given an asymmetric relation $A\subseteq V^2$ a digraph $D=(V,A)$ is minimally strong iff $D$ is strongly connected and for all arcs $\alpha\in A$ the digraph $D−\alpha=(V,A\setminus\{\alpha\})$ is ...
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93 views

Digraphs with same number of semiwalks

This is a follow-up question to Characterisation of walk-equivalent digraphs. Question: Do there exists two directed graphs $G$ and $H$ consisting of the same number ($n$) of vertices, such that \...
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1answer
105 views

Characterisation of walk-equivalent digraphs

Setting Let $G=(V,E)$ be an undirected graph. A walk $\pi$ in $G$ of length $k$ is a sequence of $k+1$ vertices $v_1,\ldots,v_{k+1}$ such that for each $i\in[1,k]$, $\{v_i,v_{i+1}\}\in E$. Let $H=(W,F)...
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1answer
100 views

Two cospectral (normal) digraphs which are not orthogonal similar

Preliminaries A complex matrix $A$ is normal when $A$ and $A^*$ commute. A real matrix $A$ is normal when $A$ and $A^t$ commute. Two complex matrices $A$ and $B$ are said to be unitary similar if ...
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1answer
220 views

Significance of the Eigenvalues of the adjacency matrix of a weighted di-graph

I'm currently running a simulation on a bunch of randomly generated points, each with two randomly selected 'partners' from the set of points. In the simulation the points try to move such that they ...
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102 views

The properties of almost all directed graphs

A mathematician on the forum previously requested a reference on human brains modelled as directed graphs. This makes sense as neurons are mostly unidirectional and I have been thinking about similar ...
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170 views

Frobenius normal form of a doubly stochastic matrix

If $A \in M_n(\mathbb{C})$, then $A$ is called reducible if there is a permuation matrix $P$ such that $$ P^\top A P = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix}, $$ in ...
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1answer
132 views

Latent Dirichlet allocation and properties of digamma function

In the paper Blei, D. M., Ng, A. Y., & Jordan, M. I. (2003). Latent Dirichlet Allocation. Journal of Machine Learning Research, 3(4–5), 993–1022. http://www.jmlr.org/papers/volume3/blei03a/blei03a....
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65 views

Proof of the asymptotic expression for the number of self-converse digraphs?

The following expression was mentioned in the master thesis of Alastair Farrugia on Page 199 of his thesis Self-complementary graphs and generalisations: a comprehensive reference manual, M.Sc. Thesis,...
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Groups that can occur as graph automorphisms of a fixed size graph

From theorem $4$ and corollary $1$ in this book we have that graph isomorphism has to do with automorphism group of a graph. We also know every group is the automorphism group of a graph by Frucht's ...
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1answer
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Finite sequences realizable by degree difference in digraphs

Let $n>0$ be an integer, and let $[n] = \{1,\ldots,n\}$. A function $f:[n]\to \mathbb{Z}$ is said to be in- and out-degree-realizable (or io-realizable for short) if there is a directed graph $G = (...
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Non-equivalent eulerian trails in $K_{2n+1}$

Two eulerian trails of $K_{2n+1}$ are defined to be equivalent if the orientations obtained by orienting the edges as traversed by the trails are isomorphic as digraph. How many non-equivalent trails ...
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1answer
79 views

Algorithm generating digraphs

Is there an algorightm generating all digraphs with $n$ edges up to isomorphism whose underlying graph is not a tree? For example, for $n=3$, there are only two such digraphs, representable as $\text{...
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428 views

Algorithms for rooted directed acyclic graph isomorphism

Given two directed acyclic graphs $G_1$ and $G_2$, and their roots $r_1$ and $r_2$, is there a polynomial algorithm to determine if $G_1$ and $G_2$ are isomorphic?
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Does every directed graph have a directed coloring with $4$ colors?

Every finite directed graph has a majority coloring with $4$ colors. (The notion of majority coloring is defined below.) Question. Can every infinite directed graph be majority-colored with $4$ ...
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Decreasing the directed chromatic number of a digraph by adding an edge

The chromatic number of an undirected graph can never be decreased by adding an edge. However, things are not that clear when we deal with coloring directed graphs - but first, the definition of this ...
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89 views

Name for Directed Edges in Digraphs

Graph theory originated in German speaking countries and there directed edges are called "Pfeil" which translates to "arrow", which makes sense, because arrows have distinguishable ...
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1answer
85 views

Generalized digraph homomorphisms and graph cores

Given any digraphs $G$ and $H$ we say a surjection $f:V(G)\to V(H)$ reduces $G$ to $H$ if and only if it satisfies $(u,v)\in E(G)\iff (f(u),f(v))\in E(H)$. Where if there exists at least one ...
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203 views

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 in $\Gamma$ ...
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2answers
256 views

Orientability of $\mathbb{Z}^n$

For any set $X$ set $[X]^2 = \{\{x,y\}: x,y\in X, x\neq y\}$. If $n\geq 2$ is an integer, we endow $\mathbb{Z}$ with a graph structure in the following way. If $x,y\in \mathbb{Z}^n$ we say $x,y$ are ...
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an inverse semigroup (and perhaps a $C^*\!$-algebra) associated with a directed graph

The following inverse semigroup associated to a directed graph came up in my research. I've read that from an inverse semigroup one may derive a $C^*\!$-algebra whose generators are partial isometries ...
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105 views

In the context of directed graphs is it standard notation to allow an element of an independent vertex set to be contained in a loop?

Given any relation $R$, that is, any set of ordered pairs, we can associate a unique digraph $D$ to our relation $R$ by setting $D=(\text{fld}(R),R)$ where $\text{fld}(R)=\text{dom}(R)\cup\text{rng}(R)...
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169 views

Similarities between isomorphism classes of homeomorphic directed graphs

To clarify, I'm speaking of homeomorphisms in a graph theoretic context, defined by subdivisions of arcs in a directed graph. A subdivision of an arc $(x,z)$ in a directed graph is obtained by ...
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55 views

Orientations in connected bridgeless graphs

Let $n\geq 3$ be an integer, set $[n] = \{1,\ldots,n\}$ and let $G=([n],E)$ be an undirected connected bridgeless graph. Is there an orientation (explanation below) of $G$ such that for all $a\neq b\...
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1answer
455 views

Tournaments with exactly one directed Hamiltonian path

Every tournament contains a directed Hamiltonian path (a path visiting every vertex exactly once). Suppose that $T$ is a tournament on $[n]:=\{1,\ldots,n\}$ for some integer $n\geq 2$ with exactly ...