Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,451
questions
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59 views
Is there a term for 'no-turn-back walk' in graph theory?
Let $G$ be a finite undirected graph. A walk in $G$ is a finite sequence $<v_1,e_1,v_2,e_2,\dots,v_{k-1},e_{k-1},v_k>$ where $v_j$'s are vertices in $G$, $e_j$'s are edges in $G$, and $e_j=v_jv_{...
1
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0answers
35 views
Does high connectivity of line graph of $G$ imply high (cyclic) connectivity of $G$?
All graph considered here are finite, simple and undirected.
We know that a graph $G$ is $k$-edge connected if and only if its line graph is $k$-connected (where $k\in\mathbb{N}$). In particular, if $...
2
votes
1answer
70 views
On cubic planar graphs with face boundaries of length divisible by 4
All graphs considered here are finite, simple and undirected.
Let $\mathscr{G}$ denote the class of cubic plane graphs for which all face boundaries are of length divisible by four. The 3-cube $Q_3$ ...
2
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0answers
56 views
Summing over weighted paths optimally
Given an edge-weighted directed graph, how do you sum over all weighted paths between A and B while using the smallest number of multiplications?
Is there a name for this problem?
This comes up in ...
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votes
1answer
311 views
Find research partner (profession and beginner)
I've 10 years of industrial work, but in my free time, I do research, write papers to conferences, help to teach to my old friend at the university and I even did a Ph.D. full-time program.
Now, I've ...
1
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1answer
168 views
How to show that Color Tiles is NP-Complete
Color Tiles is a puzzle game where color tiles are laid out in a rectangular grid. A tile is visible to an empty grid cells if there is a clear line-of-sight in one of the 4 cardinal directions (You ...
6
votes
1answer
278 views
Complexity of optimal elimination for a planar tensor network
Edit Dec 15 it's not obvious this problem is tractable when further restricting to trees, see cs.SE question
Suppose we need to sum out variables in a tensor network (a factor graph where each ...
1
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0answers
39 views
Cycle double covers of cubic graphs using only a few cycles
This is a reference request question. Let $G$ be an arbitrary cubic graph.
Is the problem of finding a cycle double cover $D$ of $G$ with minimum number of cycles in $D$ studied in the literature?
I ...
4
votes
0answers
289 views
Computational Complexity of cycle double cover
Let $\mathcal{G}$ be the set of all finite simple graphs. Let graph $G\in \mathcal{G}$ and $C_G=\left <C_1,...,C_m \right >$ be a sequence of cycles of $G$ for some $m$. For every edge $e$ of $G$...
35
votes
4answers
2k views
Why is "topological sorting" topological?
Why is "topological sorting" called "topological"? Is it just because it determines an order without altering any vertices or edges -- like a doughnut and coffee cup are topologically equivalent? Why ...
2
votes
1answer
54 views
Is every 4-colourful Eulerian Orientation of a planar 4-regular graph good?
Let $G$ be a simple undirected plane 4-regular graph. Basic definitions are given at the bottom of this question.
An Eulerian orientation of $G$ is good if for each vertex $v$ of $G$, the edges around ...
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0answers
26 views
Cycle decompositions of locally linear 4-regular graphs
(Preface)
We consider only finite, simple, undirected graphs here. An orientation of a graph $G$ is obtained by assigning some direction to each edge of $G$.
(Question starts)
A graph is locally ...
1
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0answers
41 views
Properties of half-way locally bijective homomorphisms between Eulerian orientations
Short Version
Let $G$ and $H$ be two Eulerian graphs and let $\overrightarrow{G}$ and $\overrightarrow{H}$ be Eulerian orientations of those graphs. Let $f$ be a homomorphism from $G$ to $H$.
(...
0
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1answer
65 views
Is a grid graph a vertex-minor of a complete graph? [closed]
Consider a graph $G$. A graph $H$ is the vertex-minor of the graph $G$ if $H$ can be obtained from $G$ using vertex deletions and local complementations. For more information, look at Definition 2.1 ...
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0answers
20 views
How does symmetric difference of b-matchings look like?
It is easy to see that symmetric difference of two matchings are cycles or simple paths. But what about b-matchings? Is there anything known about how they look? Even for restricted cases such as ...
8
votes
0answers
107 views
improved analysis of spectral gap of zigzag product?
I am reading the paper introducing zigzag products of expander graphs (https://arxiv.org/abs/math/0406038). The paper mentions the following observation in the introduction:
Moreover, the variational ...
1
vote
0answers
33 views
Is arrangement-type graph on cyclic $k$-permutations of $n$ already studied?
The arrangement graph $A_{n,k}$ is the graph whose vertices are $k$-permutations of an $n$-vertex set $X$ (say, $X=\mathbb{Z}_n$) and two $k$-permutations are adjacent if they differ in exactly one ...
0
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0answers
85 views
optimization on graph edges selection
I have the below problem. I wonder if there exists a similar known class of problems (e.g., in optimization, graph theory) which I can relate my problem to, and find a similar solution there.
I am ...
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1answer
103 views
Algorithm for finding traffic equilibrium
I watched a youtube video about a certain interesting property of springs and road networks. It made me think: if we represent a network of roads as a graph where edges are roads described by a ...
-1
votes
1answer
238 views
Does such a graph exist? [closed]
[EDITED FOR CLARITY]
Does there exist an edge-colored graph $G$ with the following properties?
$G$ has a vertex $r$ with exactly three, distinctly colored, incident edges: $(r, u)$, $(r, v)$, $(r, w)$...
7
votes
1answer
134 views
Monotone circuit representations of paths in a graph?
Consider a directed graph $G = (V, E)$ with a source $s \in V$ and sink $t \in V$. From $G$, I can define a monotone Boolean function $\phi_G$ on the set of variables $E$, in the following way: every ...
2
votes
1answer
197 views
Does abundance of max cliques make it easy to solve COLORABILITY?
Let $q\geq 3$. We know that $q$-COLORABILITY is an NP-complete problem.
Suppose that $G$ is a graph such that each vertex of $G$ is part of a $q$-clique (i.e. $K_q$). Since we may assume that $G$ does ...
3
votes
2answers
171 views
exact path cover for undirected graph
In a Python plotting application,
I have an undirected connected graph, not necessarily simple, that I'd like to cover with paths such that each edge is contained in exactly one path.
The number of ...
2
votes
0answers
55 views
Complexity of a matrix partition problem in graphs
All graphs in this question are finite, simple, and undirected. Let $H$ be a regular graph on at least five vertices, let $v_1,v_2,\dots,v_n$ be the vertices in $H$, and let $M$ be the adjacency ...
23
votes
0answers
1k views
$\Delta = 57, d=2$ Moore Graph
I am looking into the last open question regarding the existence of Moore Graphs of diameter 2. A problem that has been open in combinatorics for more than 55 years.
You may recall that Hoffman and ...
2
votes
1answer
99 views
Can this special case of Node Weighted Steiner Tree be solved in polynomial time?
Consider the node-weighted steiner problem:
Input: a graph $G=(V,E)$, a set $T\subseteq V$ of terminals, a weight function $w: V\setminus T \to \mathbb{R}_+$.
Output: a minimum weight subset $S \...
1
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0answers
44 views
Approximate solution for maximum coverage problem with choice constraint
Suppose a sequence of sets $S_1,S_2,...,S_i$ where each set contains sets of elements. That is, each set $S$ contains many sets $a_1,a_2,...,a_{|S|}$. We are given an integer $k$ and we assume that $\...
0
votes
0answers
21 views
Listing colourful Eulerian orientations in poly. time
An orientation $\overrightarrow{G}$ of an undirected graph $G$ is a directed graph obtained by assigning some direction on every edge of $G$.
An orientation $\overrightarrow{G}$ is said to be an ...
4
votes
1answer
168 views
Does a graph resulting from the union of triangles has a particular name?
I have different simple triangle graphs. As an instance, $G_1=(V_1,E_1)=(\{1,2,3\},\{\{1,2\},\{2,3\},\{3,1\}\})$ and $G_2=(V_2,E_2)=(\{1,4,5\},\{\{1,4\},\{4,5\},\{5,1\}\})$.
The union of both graphs ...
0
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0answers
37 views
Is there a primal-dual algorithm for the Tree Augmentation Problem or the Cactus Augmentation Problem?
The TAP problem and the CacAP problem can be seen as covering problems for the minimum cuts of a graph.
It seems like these problems would fall under the framework of network design problems (...
2
votes
0answers
64 views
What is the expected treewidth of a large-treewidth graph intersected with Erdos-Renyi graph?
Suppose we have a graph $G$ with treewidth $t$. Let $p \in (0,1)$ be a constant. Then let's independently remove each edge from $G$ with probability $p$. What is the expected treewidth of the ...
3
votes
1answer
169 views
Are there an algorithm that find Minimum spanning tree in $O(n^2\log\log^*n)$?
Given completed metric weighted graph $G=(V,E)$ that have $n$ vertices. Are there an algorithm that find MST of $G$ in $O(n^2)$?
I read abstract of this paper that mentioned an algorithm with running ...
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votes
2answers
282 views
Partition a graph into two clusters
Suppose given a complete weighted graph $G=(V,E)$, with positive weight. Are there an algorithm that partition $G$ into two clusters $C_1,C_2$ such that sum of heaviest edges in $C_1,C_2$ minimized?
...
0
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0answers
35 views
Bounded-Frequency Minimum Set Cover Problem
Consider the special case of the minimum set cover problem where each element of the universe occurs in at most 3 sets.
Can this problem be solved in polynomial time?
Is there a nontrivial upper ...
1
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1answer
175 views
State of the art approximation algorithm for $\text{MAXCUT}$ that does better than Goemans and Williamson
I had thought that the Goemans-Williamson approximation algorithm was the best for MAXCUT. To quote from Wikipedia:
The polynomial-time approximation algorithm for Max-Cut with the best
known ...
4
votes
1answer
302 views
Pagerank in directed *acyclic* graphs (DAG)
I deal with pagerank computations on large directed acyclic graphs (DAG).
I found no reference to work on this specific case, only some work on pagerank in more specific cases, e.g., PageRank of Scale ...
0
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0answers
49 views
Minimal lexicographical path on DAG in O(||V| + |E|)
Let's assume, that we have directed asyclic graph and nodes U and V. Every edge of this graph is marked with alphabet letter (alphabet size is fixed). Is there any way to answer, what is the shortest ...
5
votes
1answer
223 views
Computational complexity of Turán-type problems
According to Turán's theorem (with $r=n/2$), any graph $G$ with $n$ vertices and at least $n(n-2)/2$ edges must contain a clique of size $n/2+1$. My question is: how hard is it to find this clique?$^*$...
2
votes
0answers
93 views
How typical are odd-H-minor free graphs?
Can anything be said about how typical are odd-H-minor free graphs? (definition of odd-minor-free is in Section 2.2 of notes, page 20 of slides). For instance, for a random graph with $n$ vertices, $...
0
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0answers
21 views
Going from one base packing to another using basis exchanges
Suppose I have a matroid $M = (E, \mathcal{I})$. It is a known fact that given any two bases $X_0$ and $X_n$, we can transform $X_0$ into $X_n$ by repeatedly applying the basis exchange axiom. So ...
5
votes
2answers
257 views
Complexity of "can we get a cycle by stacking directed bipartite graphs?"
Preliminaries
We consider directed bipartite graphs of the form $G = (V,V',E)$, in which the nodes are partitioned into $V = \{1,\ldots,n\}$ and $V'=\{1',\ldots,n'\}$, with $|V|=|V'|=n$, and $E\...
6
votes
0answers
169 views
Fastest exact algorithm for MAXCUT
Is the algorithm introduced in the following paper still the fastest exact algorithm for general MAXCUT problems? TIA
Ryan Williams, A new algorithm for optimal $2$-constraint satisfaction and its ...
1
vote
1answer
99 views
Partition the edges of a bipartite graph into perfect $b$-matchings
Any $r$-regular bipartite graph can be partitioned into $r$ disjoint perfect matchings.
I want to know whether a version of this extends to perfect $b$-matchings.
Suppose we have a bipartite graph $G =...
0
votes
1answer
96 views
Max-k-cut with negative edge weights
Let $G=(V,E,w)$ be a graph and for edge $e\in E$, there is associated weight $w_e$.
The max-k-cut wants to partition V into k subsets $P_1,\cdots,P_k$ and maximize $\sum_{1\leq r<s\leq k}\sum_{i\in ...
1
vote
1answer
49 views
Graph classes where giving a q-clique edge cover makes testing for q-colouring easy
A $q$-clique of a graph is a complete subgraph on $q$ vertices.
A $q$-clique edge cover of $G$ is a set of subgraphs of $G$ such that each subgraph is a $q$-clique and each edge of G is contained in ...
1
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0answers
44 views
Prune length distribution of random binary tree
Consider a random binary tree with $N$ leaves. Each node (except the root node) has a degree of exactly three (two children and one parent). No further restriction is placed on the structure of the ...
3
votes
1answer
100 views
Finding the single-crossing embedding of a single-crossing graph
Is it known how to find a (piecewise) straight-line embedding of a single-crossing graph on the plane with exactly one crossing in polynomial time? We are currently trying to come up with a method for ...
2
votes
1answer
154 views
Coloring intersection graph of squares
It is known that the coloring intersection graph of axis-parallel rectangles is NP-Hard.
What about squares and more specific case "unit squares"?
Thanks.
0
votes
1answer
50 views
Planar 4-regular vertex-transitive graphs as system of circles
It is known that every planar 4-regular 3-connected graph $G$ admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles such that the vertices of $G$ ...
1
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0answers
142 views
How can I find the PhD thesis of A. V. Kostochka?
I've searched for the doctoral thesis of Alexandr V Kostochka in internet but couldn't find it. Can somebody help me?
I have searched in his publications list (which contains only one article ...