All Questions
Tagged with graph-theory directed-graphs
90 questions
0
votes
0
answers
19
views
2-regular directed graphs where the commutative property or relation holds at every vertex and abelian Cayley digraphs
2-regular directed graphs where the commutative property or relation holds at every vertex and abelian Cayley digraphs.
You are given a 2 regular (2-in 2-out) directed graph where you can check that ...
- 41
1
vote
0
answers
60
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 $G$ ...
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
1
answer
228
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
145
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
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
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
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
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
107
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
6
votes
0
answers
218
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
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
273
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
3
votes
1
answer
374
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
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
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
3
votes
0
answers
346
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
2
votes
0
answers
203
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-...
- 121
1
vote
1
answer
96
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?
- 48.9k
1
vote
1
answer
216
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 ...
user151154
10
votes
1
answer
370
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 ...
- 2,506
4
votes
2
answers
239
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 $1<K<P$, ...
4
votes
1
answer
208
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\...
- 2,506
1
vote
2
answers
1k
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 ...
- 21
10
votes
0
answers
438
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 ...
- 2,506
1
vote
0
answers
112
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
\...
- 493
1
vote
1
answer
138
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)...
- 493
2
votes
1
answer
122
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 ...
- 493
1
vote
1
answer
1k
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 ...
- 13
6
votes
0
answers
116
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 ...
- 3,871
3
votes
1
answer
283
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....
- 191