# All Questions

Tagged with graph-theory computational-complexity

148 questions

**1**

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34 views

### Succinct circuits and NEXPTIME-complete problems

I am fascinated by a recent fact I was reading: Succinct Circuits are simple machines used to descibe graphs in exponentially less space, which leads to the downside that solving a problem on that ...

**3**

votes

**1**answer

117 views

### Representation of Subgraph Counts using Polynomial of Adjacency Matrix

We consider a graph $G$ of size $d$ with adjacency matrix $A$, whose entries take value in $\{0,1\}$. We are interested in the number of a certain connected subgraph $S$ of size $k$ in $G$. For ...

**-4**

votes

**1**answer

195 views

### What is the computationally simplest way to universally index the set of simple graphs?

If given a simple, integer-labeled, but not necessarily connected, graph $G := (V,E)$ consisting of at least one vertex, i.e. $\lvert \rvert V \lvert \rvert \geq 1$, then is there a function to ...

**1**

vote

**1**answer

58 views

### The complexity on calculation of the Graev metric on the free Boolean group of a metric space

For a set $X$ by $B(X)$ we denote the family of all finite subsets of $X$ endowed with the operation $\oplus$ of symmetric difference. This operation turns $B(X)$ into a Boolean group, which can be ...

**2**

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**0**answers

29 views

### Flat or linkless embeddings of graph with fixed projection

The problem of finding whether a given planar diagram of a graph, with over- and under-crossings, is a linkless embedding or not has unknown complexity (Kawarabayashi et al., 2010). My first question ...

**0**

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**1**answer

82 views

### Is there any solution that currently exists for the graph automorphism problem in the general case?

I was reading the Wikipedia pages on the graph automorphism, but I could not find any solution to the problem (Not even a brute force one). So, is it indeed true that no solutions exist for the ...

**1**

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**0**answers

31 views

### Is there an algorithm for this constrained Hypergraph optimization problem?

I'm currently developing an algorithm for computing knot coloring invariants and got to the following question:
Given a set $S$ and a certain hyper-graph $H \subseteq S^3 $, find a decomposition $S = ...

**3**

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**0**answers

45 views

### Karp hardness of two cycles which lengths differ by one

Our problem is as follows:
NEARLY-EQUAL-CYCLE-PAIR
Input: An undirected graph $G(V,E)$
Output: YES if there exists $2$ (simple) cycles in $G$ which lengths differ by $1$, otherwise NO
Is ...

**6**

votes

**1**answer

107 views

### What is the complexity of counting Hamiltonian cycles of a graph?

Since deciding whether a graph contains a Hamiltonian cycle is $NP$-complete, the counting problem which counts the number of such cycles of a graph is $NP$-hard.
Is it also $PP$-hard in the sense ...

**6**

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104 views

### Does the problem of recognizing 3DORG-graphs have polynomial complexity?

A 2DORG is the intersection graph of a finite family of rays directed $\to$ or $\uparrow$ in the plane. Such graphs can be recognized effectively (Felsner et al.). A 3DORG is the intersection graph of ...

**6**

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85 views

### Combinatorial region-halfplane incidence structures

I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate.
Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\...

**3**

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**0**answers

102 views

### $\mathrm{NP}$-complete problems in graph theory: undirected vs. directed

Is it true that it is much easier to establish $\mathrm{NP}$-complete on undirected graphs than digraphs (directed graph)?
Academic articles proving $\mathrm{NP}$-completeness of problems on ...

**4**

votes

**2**answers

95 views

### Is there an efficient way to represent all non-simple cycles of a digraph up to the number of vertices?

Given two digraphs $G$ and $H$, I want a method for creating a bijection between all non-simple cycles of for all $n \le |V(G)|$. That means, given $C_G(n)$ and $C_H(n)$ being the sets of all non-...

**1**

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66 views

### Bipartite clustering is NP-hard?

Let $G = (A\cup B, E)$ be a bipartite graph with edge weights $w: E\to \mathbb{R}$. Find a partition $B_1, B_2$ of $B$ and a nonempty disjoint subsets $A_1, A_2$ of $A$ such that $w(A_1,B_1) + w(A_2, ...

**16**

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263 views

### Straight-line drawing of regular polyhedra

Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...