All Questions
33 questions
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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 ...
- 4,713
3
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2
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2k
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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
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2
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198
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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 $...
- 109k
1
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1
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182
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Lower bound on outdegree/indegree in oriented graph to guarantee cycle of length at least $k$
An oriented graph is a digraph without any self-loops, multiple arcs, or 2-cycles. What is the smallest minimum outdegree of an oriented graph on $n$ vertices that ensures there will always be a cycle ...
- 32.1k
2
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1
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228
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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 ...
- 24.2k
3
votes
1
answer
173
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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 ...
- 265
0
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51
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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
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145
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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 ...
- 1,446
8
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459
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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 ...
- 32.1k
15
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3
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613
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Maximum matching in a graph with no "shortcuts"
For a directed acyclic graph (DAG) $G$, denote by $G^\star$ the undirected graph obtained from $G$ by ignoring direction of its arcs. Let $e(G)=e(G^\star)$ be the number of arcs in $G$ (or edges in $G^...
- 109k
1
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1
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110
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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 ...
- 32.1k
3
votes
1
answer
108
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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
1
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1
answer
216
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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 ...
CommunityBot
- 1
1
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0
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65
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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
3
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1
answer
374
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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 (...
- 19.6k
2
votes
1
answer
156
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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 ...
- 1,701
4
votes
1
answer
273
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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 ...
- 7,836
6
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130
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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) &...
- 184
1
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82
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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 ...
- 4,713
3
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2
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233
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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 ...
- 3,446
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108
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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 ...
- 2,089
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438
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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
3
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0
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346
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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 ...
- 265
10
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1
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370
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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
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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 $1<K<P$, ...
- 4,589
6
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1
answer
132
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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 ...
- 37.7k
3
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2
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271
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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 ...
- 4,280
6
votes
1
answer
124
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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 — ...
- 2,966
5
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1
answer
1k
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How many hamiltonian cycles can be removed from a complete directed graph before it becomes disconnected?
The question started from a problem brought home by a friend's 5th grader: "How many ways can you seat 5 people around a round table so that the people sitting to the left of any person is different ...
- 32.1k
7
votes
2
answers
324
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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 ...
- 37.7k
1
vote
0
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108
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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, ...
- 19.6k
2
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0
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169
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When does the induced directed graph of a directed multigraph preserve information?
Let G be a directed multigraph, and let H be the induced directed graph whose vertices are the edges of G, and whose edges are given by pairs of consecutive edges in G; i.e., there is an edge from v ...
CommunityBot
- 1
5
votes
1
answer
162
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When can we make a digraph acyclic by fliping groups of arcs?
We have a digraph D=(V,A) and its arc set A is partitioned into classes. We can flip the classes, which means changing the direction of all the arcs in the class.
Is there any result on when can we ...
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