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*.
115 questions
1
vote
0
answers
56
views
Bipartite Representation of a Directed Graph
I am working on a combinatorial optimization problem and have constructed a bipartite graph as a representation of a directed graph.
The construction is as follows:
Given an initial directed graph $G$ ...
1
vote
0
answers
100
views
PageRank in directed graphs: equivalence of iterative and eigenvalue methods
Given a directed graph $ G $ with $ n $ nodes, we can represent this graph using an adjacency matrix $ A $. The stochastic matrix $ S $ can be derived from the adjacency matrix using the following ...
- 4,058
0
votes
0
answers
26
views
Expected number of nodes not involved in a cycle
Consider a random directed graph with $n$ nodes. Each node has on average $x$ number of incoming edges.
How many nodes are not part of a cycle? Alternatively phrased, how many strongly connected ...
- 101
5
votes
1
answer
386
views
Are directed graphs with out-degree exactly 2 strongly connected with probability 1?
Consider a directed graph with out-degree exactly two with $n$ vertices $v_1, v_2 \cdots v_n$ that is constructed as follows: For each vertex $v_i$, one chooses uniformly at random two (not ...
- 6,969
0
votes
0
answers
36
views
A class of directed graph, when their minimal polynomial of the adjacency matrix matches the characteristic polynomial
We consider an unweighted directed simple graph, $G$, with a Hamiltonian cycle.
Q. Assume that the adjacency matrix of $G$ is non-singular. Do the characteristic and minimal polynomials of the ...
- 4,058
1
vote
2
answers
198
views
Topology of directed graph $G$ with non-singular adjacency matrix
Given a directed graph $G$ with non-singular adjacency matrix,
Q. Is there a directed
subgraph $H$ in $G$ that can be represented as the union of disjoint cycles such that $H$ contains all nodes of $...
- 4,058
2
votes
0
answers
34
views
Do digraphs with "other" symmetries have interesting properties
Question:
do digraphs $G(V,A)$, whose adjacency matrix exhibits certain symmetries, have mathematically interesting properties?
The most famous such symmetry is $(i,j)\in A\iff(j,i)\in A$ for which ...
- 13.2k
2
votes
1
answer
227
views
Name for generalization of trees to digraphs
One definition of tree in graph theory could be as follows:
A tree is a(n undirected) graph for which there is a unique (undirected) path between any pair of vertices.
This suggest a possible ...
- 13.2k
3
votes
1
answer
173
views
Subset of the vertices in a tournament
Suppose we have a directed complete graph. Can we always find a subset $S\neq \emptyset$ of the vertices such that for every vertex $v$, $v$ has incoming edge from at least $\dfrac{|S|}{2}$ of the ...
- 169
0
votes
0
answers
51
views
Kernel perfection in some powers of cycles
Suppose I orient the edges of the power of cycle graph $G=C_n^k$ where $n=16$ and $k=4$ in such a way that all the generated cycles by the elements $1,2,3,4$ are given the standard lexical orientation....
- 2,089
1
vote
0
answers
144
views
Misunderstanding the definition of kernel in digraphs
By Borodin–Kostochka–Woodall '97 paper, the first paragraph says that directed odd cycles do not have kernels. But, I don't get this. Like, consider any $\lfloor\frac{n}{2}\rfloor$ set of independent ...
- 2,089
1
vote
0
answers
37
views
Both-way flows in a directed graph
Let $G$ be a finite directed graph, and let $s,t$ be two distinct vertices.
Problem $1(s,t)$. Find the maximum number of mutually edge-disjoint directed paths from $s$ to $t$. OK, I didn't think of ...
- 37.7k
4
votes
0
answers
317
views
What is the geometric interpretation of the first Hochschild homology group of path algebra constructed from a directed graph?
Let $\mathcal{G} = (V, E, s, t)$ is a directed graph, where $V$ - the set of its vertices, $E$ - the set of its edges, $s: E \rightarrow V, s((v_1, v_2)) = v_1$ and $t: E \rightarrow V, s((v_1, v_2)) =...
- 91
0
votes
0
answers
59
views
Maximal family of cycles in directed 3-regular graphs
Consider a finite directed 3-regular graph $G=(V,E)$ where $(v,w)\in E$ implies also that $(w,v)\in E$. I am looking for a maximal set $\mathcal{C}$ of simple cycles of length greater of equal $3$ ...
- 91
1
vote
0
answers
40
views
Probability of generating the same DAG after picking a topological ordering 3 times in a row
Consider the following process:
Choose a random permutation $p$ of $\{1, 2, \dots, n\}$ out of $n!$ options.
Choose a random directed acyclic graph $G$ that has $p$ as a topological ordering out of $...
- 1,171
8
votes
0
answers
459
views
Extension of Erdős-Gallai (s,t)-path theorem to directed graphs
The following is a result of Erdős-Gallai from 1959 (https://link.springer.com/article/10.1007/BF02024498):
Given a 2-connected undirected unweighted graph with minimum degree at least $d$, for every ...
- 81
9
votes
2
answers
363
views
Does a strong digraph always admit a vertex that lies on some path between $\Theta(n^2)$ pairs of vertices?
Let $G$ be a directed graph.
Call a vertex $v$ in $G$ central if there exists $\Theta(n^2)$ distinct pairs of vertices $(u,w)$ such that $v$ lies on some path from $u$ to $w$. We do not care whether ...
- 235
1
vote
1
answer
66
views
Path cover with sets of nodes
I am considering the following variant of the path-cover problem. I have an acyclic directed graph G=(V,E). Moreover, the set V is partitioned into $V=V_1 \cup ... \cup V_k$ (these sets are pairwise ...
1
vote
1
answer
143
views
Eigenvalues of directed graph with one outward edge for each vertex
I am concerned with unweighted directed graphs where each node contains exactly one edge pointing to another node, which could be itself. In other words, each row of the adjacency matrix contains one ...
- 283
0
votes
0
answers
119
views
Lengths of paths through Conway’s Game of Life
This question is inspired by the following challenge from CodeGolf.SE: https://codegolf.stackexchange.com/q/251510/88765.
Given positive integer $N$, we can consider a version of Conway’s game of life ...
- 3,499
3
votes
1
answer
103
views
Minimal digraph covering with no 2-path edge sets is of size $\left( 1 + o \left( 1 \right) \right) \log_2 \chi(G)$
The last problem in 2022 IMC Day 1 strongly correlates with graph theory. In its official solution, the fundamental approach can be rephrased as follows.
Give a digraph $G=(V,E)$. We call a subset of ...
1
vote
2
answers
106
views
Minimum edge-weighted directed subgraph in polynomial time
I am looking for an algorithm with polynomial complexity where, given a strongly connected edge-weighted digraph I can find the minimal subgraph which connects some root vertex v to a known set of ...
- 13
3
votes
1
answer
108
views
Digraph without "immediately isomorphic" vertices?
Say that a digraph $(V,E)$ is reducible if there exist $x,y\in V$ with $x\ne y$ and such that for all $z\in V$, $(x,z)\in E\leftrightarrow(y,z)\in E$ and $(z,x)\in E\leftrightarrow(z,y)\in E$. It is ...
- 423
2
votes
1
answer
73
views
Generate all strongly connected tournament
I want to generate all strongly connected tournament of size $n \in \{4, 11\}$.
As a strongly connected tournament has an hamiltonian path I may assume that $v_i v_{i+1}$ is always an arc, and $v_n ...
- 267
6
votes
0
answers
217
views
Maximum number of Hamilton paths in a tournament on $n$ vertices
Recall that a tournament is a directed graph $T$ such that for every pair of distinct vertices $\{v,w\}$, exactly one of the ordered pairs $(v,w)$, $(w,v)$ is an arc of $T$.
A tournament is strongly ...
- 12.7k
1
vote
0
answers
65
views
Partitioning antidirected trees with bounded degree, such that the graph induced by the partition is a constant antidirected tree
Given a partition of the vertices of a graph, we can define an auxiliary graph which conveys information about the edges between sets of the partition. This defines a graph with vertex set equal to ...
- 71
1
vote
0
answers
50
views
Reference for a lemma on acyclic subgraph
Lemma. Let $D$ be a digraph. Then there exists an acyclic subdigraph $D'$ of $D$ such that the total degree (i.e. out-degree plus in-degree) of $v$ in $D'$ is at least the out-degree of $v$ in $D$ for ...
- 121
4
votes
0
answers
59
views
Graph-class defined by matrix-like vertex-operations
Let $m$ be a positive integer. We define a (directed) graph on $m(m-1)$ vertices
$$V = \bigl\{(i,j) \mid i \ne j,\, i,j\in\{1,\dots,m\}\bigr\}$$
and edges as follows:
$(i,j) \in V$ is adjacent (...
- 577
2
votes
1
answer
136
views
Is the set of "endomorphisms" of a directed set again a directed set?
Let $(I,\leq)$ be a directed set, that is $\leq$ is reflexive and transitive and for every $a,b\in I$ we find $c\in I$ such that $a,b\leq c$. Now consider the set $M$ consisting of all maps $\sigma:I\...
- 696
1
vote
0
answers
37
views
Given a simple undirected graph, how to direct its edges to make it transitive
I am currently looking for an algorithm to determine whether we can direct every edge (no adding or deleting edges allowed) so that the graph is transitive (meaning that if (x,y) and (y,z) are edges ...
- 261
1
vote
2
answers
1k
views
Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?
I am looking at doing some basic validation on a database of st-dags. It would be useful to have:
A formula for the number of non-isomorphic st-dags with n vertices
A formula for the same with n ...
- 21
0
votes
0
answers
125
views
Are there any necessary conditions of the existence of a Hamiltonian cycle on directed graphs
I'm trying to prove that one concrete directed graph has no Hamiltonian cycle, but didn't seem to find any relevant theorems
2
votes
1
answer
156
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 ...
- 71
6
votes
0
answers
130
views
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) &...
- 184
4
votes
1
answer
272
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 ...
- 1,701
9
votes
1
answer
333
views
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 ...
- 1,689
3
votes
1
answer
373
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 (...
- 143
1
vote
0
answers
82
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 ...
- 111
3
votes
2
answers
2k
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?...
- 1,701
1
vote
0
answers
158
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 ...
- 161
4
votes
1
answer
310
views
Using a poset or directed graph as input for a neural network
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 ...
1
vote
0
answers
108
views
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 ...
- 2,089
3
votes
2
answers
233
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 ...
- 31
4
votes
1
answer
207
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 $...
- 3,446
6
votes
1
answer
610
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 ...
- 612
0
votes
1
answer
102
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.
...
1
vote
0
answers
223
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. ...
- 19
1
vote
1
answer
110
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 ...
- 130
2
votes
0
answers
32
views
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}(...
- 23
3
votes
0
answers
345
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 ...
- 265