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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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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61 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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Find maximum directed graph

For a directed graph $G=(V,E)$, we want to compute the max number $k$ such that there exists a directed graph $H=(V,E')$ such that $E'\subseteq V\times V$ and $|E'|=k$ such that For any $v\in V$, $(v,...
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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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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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42 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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44 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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129 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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206 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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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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178 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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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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382 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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Twisted Laplace eigenvalues on graph

Let $X$ be a regular finite connected graph and $\Gamma$ its fundamental group. Fix a character $\chi:\Gamma\to\mathbb{C}^\times$. This defines a locally constant sheaf on $X$ and there is an ...
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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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99 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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91 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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167 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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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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137 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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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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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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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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67 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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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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71 views

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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86 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 front end and rear ...
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81 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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255 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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101 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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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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444 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 ...
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Sum-coloring a tournament

If $G=(V,E)$ is a loopless finite directed graph and $v\in V$, we set $\text{In}(v) = \{(w,v): w\in V \land (w,v) \in E\}$. Let $T=(V,E)$ be a tournament such that for every $v\in V$ the set $\text{...
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Characterizing SP-DAGs by Forbidden Minors?

So it's well-known that an alternative way to define a series-parallel (undirected graph) is by the forbidden minor $K_4$. Is there a known analog of this definition for directed graphs — ...
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231 views

Graph isomorphism problem for minimally strongly connected digraphs

A minimally strongly connected digraph (MSC) is strongly connected (SC), while removal of any arc destroys this. That is, between any two vertices a, b there exists a directed path from a to b, while ...
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What Kind of Graph is This?

I am currently developing TSP heuristics that aim at symmetrically reducing the original, complete and undirected graph. The overarching rationale is that the reduction is done via a sequence of ...
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120 views

Length of longest directed circuit in random tournament

Build a random tournament $T=(V,E)$ on $V=\{1,\ldots, n\}$ in the following fashion: for $i < j\in \{1,\ldots, n\}$ let the probability be $0.5$ whether $(i,j)\in E$ or $(j,i)\in E$ (in a ...
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608 views

Is transitive reduction for a direct acyclic graph really unique? [closed]

According to Wikipedia, "If a given graph is a finite directed acyclic graph, its transitive reduction is unique" Here is what I think might be a counter-example: Imagine a diamond-shaped DAG where ...
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Generating tournaments inductively

This is a somewhat vague question, but I'm interested in ways to create a strong tournament from one or more smaller tournaments. Obviously, the disjoint union of two tournaments is a new tournament, ...
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813 views

Minimum negative eigenvalue of zero-one matrices

The following question must have been answered decades ago. For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...
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231 views

The minimum number of Hamiltonian paths in a strongly connected tournament of order $n$

For $n\ge3$ let $a(n)$ be the minimum number of Hamiltonian paths in a strong (i.e., strongly connected) tournament of order $n.$ Where is $a(n)$ discussed in the literature? Is the exact value ...
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2k views

Removing cycles in a directed graph by swapping edges orientation

I have the following problem: let $G$ be a finite directed graph with $V$ vertices $v_i$ and $E$ (directed) edges $e_j$. I know that if an edge $e_k = (v_i, v_j)$ is in the graph, then the opposite ...
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225 views

Name for directed graphs with “balanced cycles”

Does the following class of graphs have a name? I'm interested in directed graphs with the following property: for every cycle (of the underlying undirected graph) half of the edges go in one ...
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The number of Hamiltonian paths in a tournament

If $h(T)$ denotes the number of (directed) Hamiltonian paths in the tournament $T,$ what is the range of $h(T)$ as $T$ ranges over all (finite) tournaments $T$? By a classical theorem of Rédei [...
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224 views

Directed homotopy in the Cayley graph of a monoid

There is a the notion of the Cayley graph $C(G)$ of a group $G$ (which depends on a given presentation $G \cong \mathcal F(S) / \sigma$ where $\mathcal F$ is the free group functor and $\sigma$ some ...