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Questions tagged [graph-theory]

Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.

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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 ...
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1answer
936 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 ...
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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 ...
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3answers
2k 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 ...
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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
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4answers
746 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
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6answers
463 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
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1answer
380 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,...
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3answers
590 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
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2answers
347 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
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1answer
542 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 ...
17
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1answer
593 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 ...
15
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1answer
509 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
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6answers
607 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
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1answer
882 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 ...
35
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9answers
10k views

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, ...
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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
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3answers
900 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 ...
3
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2answers
319 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 ...
2
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1answer
335 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 ...
6
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4answers
731 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
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2answers
815 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 $(k,\epsilon)$-...
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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 ...
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3answers
1k 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 ...
23
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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. ...
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2answers
3k 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
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1answer
302 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
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1answer
340 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 (...
16
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2answers
607 views

H-free cut problem

Suppose you are given a connected, simple, undirected graph H. The H-free cut problem is defined as follows: Given a simple, undirected graph G, is there a cut (partition of vertices into two ...
20
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2answers
889 views

Succinct circuit representation of graphs

The complexity class PPAD (e.g. computing various Nash equilibria) can be defined as the set of total search problems polytime reducible to END OF THE LINE: END OF THE LINE: Given circuits S and P ...
15
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1answer
732 views

Online transitive closure better than O(N^2) per edge addition

I'm looking for an online algorithm to maintain the transitive closure of a directed acyclic graph with a time complexity less than O(N^2) per edge addition. My current algorithm is like this: ...