Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,403
questions
10
votes
1answer
945 views
Finding short and fat paths
Motivation: In standard augmenting path maxflow algorithms, the inner loop requires finding paths from source to sink in a directed, weighted graph. Theoretically, it is well-known that in order for ...
10
votes
2answers
644 views
Approximating non-trivial graph automorphism?
Graph automorphism is a permutation of graph nodes that induces a bijection on the edge set $E$. Formally, It is a permutation $f$ of nodes such $(u,v)\in E$ iff $(f(u),f(v))\in E$
Define an ...
13
votes
1answer
516 views
Is “Is a permutation p an automorphism of a graph in my set?” NP-complete?
Suppose we have a set S of graphs (finite graphs, but an infinite number of them) and a group P of permutations that acts on S.
Instance: A permutation p in P.
Question: Does there exist a graph g in ...
10
votes
4answers
503 views
Interesting functions on graphs that can be efficiently maximized.
Say that I have a weighted graph $G = (V,E,w)$ such that $w:E\rightarrow [-1,1]$ is the weighting function -- note that negative weights are allowed.
Say that $f:2^V\rightarrow \mathbb{R}$ defines a ...
14
votes
1answer
318 views
Generating Graphs with Trivial Automorphisms
I'm revising some cryptographic model. To show its inadequacy, I've devised a contrived protocol based on graph isomorphism.
It is "commonplace" (yet controversial!) to assume the existence of BPP ...
10
votes
1answer
303 views
CSPs with unbounded fractional hypertree width
At SODA 2006, Martin Grohe and D$\acute{\rm a}$niel Marx's paper "Constraint solving via fractional edge covers" (ACM citation) showed that for the class of hypergraphs $H$ with bounded fractional ...
1
vote
0answers
2k views
DAG partitioning to subgraphs
Given a DAG with $|V| = n$ and has $s$ sources, we have to present subgraphs such that each subgraph has approximately $k_1=\sqrt{s}$ sources and approximately $k_2=\sqrt{n}$ nodes.
(Note: ...
6
votes
1answer
4k views
What is the computational complexity of the PageRank problem?
I was just wondering what the complexity of the PageRank problem is. A description can be found here: http://en.wikipedia.org/wiki/PageRank . (I am referring to the problem that is solved by the ...
8
votes
1answer
703 views
Max-clique in line graph of hypergraph
Suppose we have a multigraph (later, a multihypergraph). An edge-clique is a set of edges which all pairwise intersect (have at least one common vertex). Then any edge-clique $C$ in a multigraph ...
11
votes
3answers
2k views
Extension to the Stable Marriage Problem ?
This may sound more like a social sciences question more than a TCS one, but it is not. When reading "Randomized Algorithms" which describes the Stable Marriage Problem, one can read the following (...
4
votes
0answers
224 views
Follow-up on Nair/Tetali's Correlation Decay Tree for hypergraphs?
I'm wondering if anybody has followed up on Nair/Tetali's correlation decay tree construction for hypergraphs. I didn't find anything relevant in back-citation on google scholar. Interesting question ...
6
votes
0answers
210 views
Further question on hardness of node partitioning under shortest path constraint
This question first appeared at Hardness of node partitioning under shortest path constraint and I restate it here
Given a direct graph $G=(V,E)$. $\forall (i,j) \in E$, there is a weight $w(i,j) \...
1
vote
1answer
3k views
Pre-order traversal on a search tree
The following is from this year's CS GRE practice test. I've worked through the test and I've been able to understand every question except for this one. Can anyone help me understand what's going on ...
26
votes
2answers
3k views
Hamiltonicity of k-regular graphs
It is known that it is NP-complete to test whether a Hamiltonian cycle exists in a 3-regular graph, even if it is planar (Garey, Johnson, and Tarjan, SIAM J. Comput. 1976) or bipartite (Akiyama, ...
26
votes
3answers
954 views
When is relaxed counting hard?
Suppose we relax the problem of counting proper colorings by counting weighted colorings as follows: every proper coloring gets weight 1 and every improper coloring gets weight $c^v$ where $c$ is some ...
4
votes
1answer
597 views
Hardness of node partitioning under shortest path constraint
Given a direct graph $G=(V,E)$. $\forall (i,j) \in E$, there is a weight $w(i,j) \in R$ (negative weight is possible). A label $l(i)$ is associated with each node $i \in V$. How to assign $k$ (or less)...
13
votes
1answer
4k views
What is the correct definition of $k$-tree?
As the title says, what is the correct definition of $k$-tree? There are several papers that talk about $k$-trees and partial $k$-trees as alternative definitions for graphs with bounded treewidth, ...
5
votes
1answer
223 views
How can one construct a densest graph with no k-clique?
Given integers $k$ and $n$ with $2 \le k < n$,
how does one construct a graph on $n$ vertices
that contains no $k$-clique and has the maximal
number of edges?
This sounds like basic ...
23
votes
1answer
867 views
Logspace algorithms on graphs with bounded tree width
Tree width measures how close a graph is to a tree. It is NP-hard to compute tree width. The best known approximation algorithm achieves $O(\sqrt{{\log}n})$ factor.
Courcelle's theorem states that ...
2
votes
1answer
263 views
Number of Vertex Covers and Permanent
Is there any relationship between the number of vertex covers of a graph $G$ and the permanent of $G$'s adjacency matrix?
2
votes
2answers
2k views
polygonal triangulation and 3-colorability
Lets define polygonal triangulation a triangulation which has a hamiltonian cycle.
It's easy to see that any polygonal triangulation is 3-colorable since any triangulation of a polygon is 3-colorable....
17
votes
3answers
1k views
Properties of Random Directed Graphs with Fixed Out-Degree
I am interested in properties of random directed graphs with fixed out-degree $d$. I am imagining a random graph model where each vertex chooses d neighbors (say, with replacement) u.a.r.
...
1
vote
1answer
1k views
Removing all but a few cycles in a graph
Let problem $S$ be defined as
Given undirected graph $G$ and a set
of cycles $C_1,C_2, \ldots, C_n$ in G,
find minimum number of vertices that
need to be deleted to remove all
cycles in the ...
7
votes
1answer
949 views
Graph Theory Fun Problem
Show that in any graph $G$ with min-degree $k$ ($k \geq 1$ duh!) you can find as its subgraph any tree on $k+1$ vertices.
I have not been able to solve the question so far. However, I would like if ...
-2
votes
1answer
2k views
How do I formally describe a rooted, directed, acyclic graph?
I need a formalism to describe the following requirements:
I have a graph comprised of nodes and transitions between nodes
Nodes maybe one of three types, all are sub-classes of a base abstract node ...
0
votes
3answers
4k views
Is it possible to have a 4-coloring for a non-planar graph ? [closed]
I have been working on this thread Grid $k$-coloring without monochromatic rectangles, and I am aware that the four color theorem implies that all planar graphs are four colorable.
The question is ...
7
votes
3answers
5k views
Polynomial Time Algorithm for Graph Isomorphism Testing [closed]
"Michael I. Trofimov" claims that he has found a poly-time algorithm for graph isomorphism, which works for all graphs.
The paper is given in arXiv. The companion website gives a proof-of-concept ...
7
votes
4answers
847 views
A relaxed Steiner Tree Problem
Given a weighted graph $G(V,E,w)$ where $w$ is the weight function on edges and a subset of vertices $S\subseteq Q$ called terminals, a Steiner Tree is a connected subgraph which connects all vertices ...
15
votes
6answers
491 views
Global properties of hereditary classes?
A hereditary class of structures (e.g. graphs) is one that is closed under induced substructures, or equivalently, is closed under vertex removal.
Classes of graphs that exclude a minor have nice ...
11
votes
1answer
431 views
Computation of max H-free sets
In a graph, an independent set is a vertex subset which doesn't contain an edge as an induced subgraph. The problem of finding largest independent sets in a graph is a fundamental algorithmic question,...
6
votes
3answers
762 views
In Strongly connected tournament T.Is it NP-hard to find a minimum number of vertices(V1) which make T\V1 as non strongly connected tournament.
Given strongly connected tournament T.find a minimum number of vertices(V1) which make T\V1 as non strongly connected tournament.
I have doubt whether the problem mentioned can be solved in polynomial ...
13
votes
2answers
375 views
H-free partition
This is a question inspired by the H-free cut problem. Given a graph, a partition of its vertex set $V$ into $r$ parts $V_1, V_2, \ldots, V_r$ is $H$-free if $G[V_i]$ does not induce a copy of $H$ for ...
1
vote
2answers
781 views
Does this notation have a special meaning?
I am currently reading a paper and I don't know how to interpret this notation you can see on the screenshot.
http://moxn.brainex.de/pub/dfg.png
Do the pointy angle brackets have a special meaning ...
19
votes
1answer
702 views
Rapidly mixing Markov chains on 3-colorings of a cycle
The Glauber dynamics is a Markov chain on the colorings of a graph in which at each step one attempts to recolor a randomly chosen vertex with a random color. It does not mix for the 3-colorings of a ...
16
votes
1answer
558 views
Sensitivity of Graph Properties
In [1], Turan shows that the sensitivity (called "critical complexity" in the paper) of a graph property is strictly greater than $\lfloor {1\over 4} m \rfloor$ where $m$ is the number of vertices in ...
8
votes
6answers
640 views
Have any generalizations of maximum weight matching been studied?
For example, one way to view maximum weight matching is that each vertex $v$ gets a utility $f_v= w(e_v)$ that equals the weight of the edge it's matched on, and zero otherwise.
accordingly, a ...
19
votes
1answer
1k views
Construction of graphs where every pair of vertices have an unique common neighbor
Let $G$ be a simple graph on $n$ vertices $(n > 3)$ with no vertex of degree $n − 1$. Suppose that for any two vertices of $G$, there is a unique vertex adjacent to both of them. It is an exercise ...
40
votes
10answers
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Data for testing graph algorithms
I am looking for a source of huge data sets to test some graph algorithm implemention. Please also provide some information about the type/distribution (e.g. directed/undirected, simple/not simple, ...
34
votes
3answers
2k views
Given a weighted dag, is there an O(V+E) algorithm to replace each weight with the sum of its ancestor weights?
The problem, of course, is double counting. It's easy enough to do for certain classes of DAGs = a tree, or even a serial-parallel tree. The only algorithm I have found which works on general DAGs in ...
2
votes
3answers
1k views
Best bounds for the longest path optimization problem in cubic Hamiltonian graph?
optimization problem
Input: cubic Hamiltonian graph
feasible solution: A simple path
measure to optimize: length of the simple path
Design a polynomial-time algorithm that outputs the longest path ...
asked Aug 27 '10 at 13:36
Mohammad Al-Turkistany
20.3k4 gold badges56 silver badges142 bronze badges
3
votes
2answers
321 views
What is the complexity of computing a compatible 3-coloring of a complete graph?
Given a complete graph whose edges are colored by 3 colors, a compatible 3-coloring is a coloring of nodes such that no edge of the graph has the same color as its end-points.
The best algorithm I ...
asked Aug 26 '10 at 18:58
Mohammad Al-Turkistany
20.3k4 gold badges56 silver badges142 bronze badges
2
votes
1answer
341 views
What is the most efficient algorithm to sample graphs with trivial automorphism groups ?
Let us call a graph "asymmetric" if it has no nontrivial automorphism group. http://en.wikipedia.org/wiki/Asymmetric_graph
I'm looking for an efficient way to compute a random asymmetric graph on a ...
asked Aug 26 '10 at 15:24
Mohammad Al-Turkistany
20.3k4 gold badges56 silver badges142 bronze badges
6
votes
4answers
892 views
What are the best known upper bounds and lower bounds for computing O(log n)-Clique?
Input: a graph with n nodes,
Output: A clique of size $O(\log n)$,
Providing links to references would be great
24
votes
2answers
863 views
Space efficient “industrial” unbalanced expanders
I am looking for unbalanced expanders that are "good" and "space-efficient". Specifically, a bipartite left-regular graph $G=(A,B,E)$, $|A|=n$, $|B|=m$, with left degree $d$ is a $(k,\epsilon)$-...
28
votes
3answers
1k views
How to produce a random graph that does not have a Hamiltonian cycle?
Let class A denote all the graphs of size $n$ which have a Hamiltonian cycle. It is easy to produce a random graph from this class--take $n$ isolated nodes, add a random Hamiltonian cycle and then add ...
23
votes
3answers
2k views
What bounds can be put on counting reachable nodes in a dag?
Given is a dag. You want to label each node by how many nodes are reachable from it. $O(V(V+E))$ is a trivial upper bound; $\Omega(V+E)$ is a lower bound (I think). Is there a better algorithm? Is ...
24
votes
6answers
2k views
Graph families which have polynomial time algorithms for computing the chromatic number
Post updated on 31st of August: I added a summary of the current answers below the original question. Thanks for all the interesting answers! Of course, everyone can continue posting any new findings. ...
11
votes
2answers
4k views
Any fast algorithm for minimum cost feedback arc set problem?
In a directed graph, $G=(V,E)$, $F\subset E$, if $G\setminus F$ is a DAG(directed acyclic graph), $F$ is called a feedback arc set.
If each edge is associated with a weight $w$, the minimum cost ...
13
votes
1answer
334 views
Finding odd holes in circulant Paley graphs
The Paley graphs Pq are those whose vertex-set is given by the finite field GF(q), for prime powers q≡1 (mod 4), and where two vertices are adjacent if and only if they differ by a2 ...
23
votes
1answer
368 views
Cliquewidth of Almost Cographs
(I posted this question to MathOverflow two weeks ago, but so far without a rigorous answer)
I have a question about graph width measures of undirected simple graphs. It is well-known that cographs (...
