# Questions tagged [graph-theory]

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

1,231 questions
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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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 (...
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 ...