# Tagged Questions

**2**

votes

**0**answers

125 views

### NP-hard proof of optimization version of exact cover [closed]

Exact cover is NPC.
http://en.wikipedia.org/wiki/Exact_cover#Equivalent_problems
Given a collection $\mathcal{S}$ of subsets of a set $X$, an exact cover is a >>subcollection $\mathcal{S}^*$ ...

**2**

votes

**0**answers

166 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**2**

votes

**1**answer

217 views

### How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem).
Restricted GA
Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...

**0**

votes

**0**answers

43 views

### Complexity of graph isomorphism in $(P_4 \cup K_1,\overline{3K_2})$-free graphs

Related to this question where isomorphism preserving
transformation maps triangle-free graphs to $(P_4 \cup K_1,\overline{3K_2})$-free graphs.
What is the complexity of graph isomorphism in $(P_4 ...

**1**

vote

**0**answers

52 views

### Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs

Let $C$ be a graph class defined by a finite
number of forbidden induced subgraphs, all
of which are cyclic (contain at least one cycle).
Are there graph problems that can be solved in
...

**34**

votes

**6**answers

3k views

### Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...

**13**

votes

**1**answer

604 views

### Who first dubbed them “expander graphs”?

Expander graphs
("sparse graphs that have strong connectivity properties")
burst onto the mathematical scene around the millennium, but I have not
been successful in tracing the origin of
(a) the ...

**0**

votes

**0**answers

125 views

### Basis of periodic tiling of Wang tile

Given a set of Wang tile,
Given 3 periodic tiling: A, B, C
We define 3 vector F[A], F[B], F[C]
each vector correspond to the appearing frequency of each type of tiles in the tiling.
Now, we ...

**1**

vote

**1**answer

90 views

### simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...

**1**

vote

**0**answers

104 views

### Graph theoretical representation of Wang Tile

We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution.
However, is there a well established counter-part ...

**1**

vote

**1**answer

81 views

### Is undirected short-simple-path-through-3-vertices decidable in polynomial time?

Consider the following language:
$L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$.
($G$ ...

**3**

votes

**1**answer

139 views

### Is the domination number NP for non-bipartite graphs?

Calculating the domination number is an NP-Hard problem. Does it remain NP-Hard if we restrict it to non-bipartite graphs?

**7**

votes

**1**answer

406 views

### Seeming contradiction about P vs NP between graphclasses.org and at least two papers about clique in even-hole-free graphs

I believe correctness about clique in even-hole-free graphs
of graphclasses.org
and the paper Vertex elimination orderings for hereditary graph classes, Pierre Aboulker, Pierre Charbit, Nicolas ...

**8**

votes

**1**answer

327 views

### A combinatorial problem concerned with logic circuits

Consider a logic circuit with two-bit gates only. The length of each gate is the number of bit lines that the gate crosses. How hard is to compute the maximum length for a given circuit? Notice that ...

**3**

votes

**2**answers

200 views

### Making a graph claw-free by adding as few edges as possible

Independent set is polynomial in claw-free graphs,
so I am wondering if this can approximate independent set.
By adding enough edges to $G$ and gets claw-free $G'$.
IS in $G'$ is IS in $G$, so this ...

**4**

votes

**2**answers

223 views

### Cubic graphs decompositions

There are many interesting computational problems related to connected cubic graph decomposition. For instance, decomposition of cubic graph into a perfect matching and a connected 2-factor ...

**3**

votes

**0**answers

127 views

### Partitioning a cubic graph into two induced cycles of equal order

I am aware that deciding the existence of a partition of the vertices of a connected graph $G(V, E)$ into two induced cycles is $NP$-complete(Theorem 2). Induced cycle is a cycle without any chord ...

**1**

vote

**1**answer

153 views

### Graph classes where Hamiltonian Cycle and Hamiltonian Path problems have different complexity

While searching The information System on Graph Classes and their Inclusions, I stumbled on several graph classes for which the Hamiltonian Cycle problem is $NP$-complete while the complexity of ...

**5**

votes

**0**answers

113 views

### Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...

**3**

votes

**1**answer

175 views

### Minimum distance between Hamiltonian cycles in cubic Hamiltonian graph

It is $NP$-hard to find constant factor approximation of longest cycle in cubic Hamiltonian graphs. Therefore, finding a Hamiltonian cycle in a cubic Hamiltonian graph is NP-hard.
By Smith's theorem, ...

**3**

votes

**1**answer

165 views

### Hamiltonian circuit

Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior.
Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...

**0**

votes

**0**answers

68 views

### maximum weight k-edge problem

Given positive integer $k$ and an undirected graph $(V,E)$, with nonnegative (non-uniform) weights on the nodes. Find $k$ edges whose spanning nodes have the maximum weight.
Is this in P or NP? I ...

**5**

votes

**5**answers

303 views

### Efficient Hamiltonian cycle algorithms for graph classes

Generally speaking finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $G$ then we can reduce the finding of a Hamiltonian cycle in $G$ to a Eurler your of $H$ ...

**4**

votes

**1**answer

155 views

### NP-hardness of sparsest cut

Consider bipartitioning the vertices of a graph $(V,E)$ into $V = P \cup Q$ to minimize $$\frac{|E(P,Q)|}{|P| |Q|},$$ where $E(P,Q)$ denotes the set of edges in the cut. The usual citation for ...

**2**

votes

**0**answers

55 views

### Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...

**1**

vote

**0**answers

92 views

### Schönhage's SMM with only one instruction

It is possible to implement $\lambda$-calculus in SchĂ¶nhage's storage modification machine using an infinite set of nodes and one single program consisted exclusively of (about hundred) instructions ...

**1**

vote

**1**answer

107 views

### Separation of Anti-Hole Inequality

Given an undirected graph $G=(V,E)$ with no loops or multiple edges, a stable set is a set of vertices for which no two vertices are adjacent.
An induced subgraph $H$ of $G$ is called an odd-antihole ...

**1**

vote

**2**answers

111 views

### complexity of finding optimal matchings of given fixed size

It is known, that maximal matchings (i.e. matchings with the maximal number of edges) and optimal matchings (i.e. matchings for which the sum of edge weights is optimal) can be calculated in ...

**6**

votes

**1**answer

236 views

### Do sparse DAGs can have large min-cuts?

For a graph $G$, let $e(G)$ denote the number of its edges, and $c_k(G)$ the smallest number
of edges that must be removed in order to destroy all paths of length $\geq k+1$.
Note that $c_1(G)\geq ...

**1**

vote

**0**answers

129 views

### NP problem implications [closed]

Hi, i would like to have some clarification on NP-completeness.
In particular I'm reading an article where they show:
1) Partitioning the edges of a graph into connected component of 3 edges (3-path ...

**4**

votes

**1**answer

226 views

### If a graph invariant is NP-Hard, is its “deck ratio” NP-Hard as well?

This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define ...

**1**

vote

**2**answers

131 views

### Is number of quasi-kernels NP-hard?

A quasi-kernel in a directed graph D is an independent subset of vertices $S$ so that for every $v \in V(D)-S$ either $v->s$ for some $s \in S$ or $v->w->s$ for some $w \in V(D)-S, s \in S$.
...

**2**

votes

**0**answers

176 views

### A natural problem on “cartesian union” of set families (hypergraphs). Does anybody know NP-complete problems related to the notion of cartesian product?

I'm curious whether the problem below is NP-complete.
I provide two simple definitions and one example at first.
Definition 1.
Let $\langle {\cal{S}}_i\rangle\substack{i\in I}$ and $\langle ...

**3**

votes

**2**answers

1k views

### Generation of All Path in a Directed Acyclic Graph

I am working on a very large dataset of a single DAG whose vertices have a low branching factor. I need to generate all possible (simple) paths starting from the source and write them to a file.
My ...

**10**

votes

**1**answer

634 views

### A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time.
...

**0**

votes

**1**answer

187 views

### graph to tree and graph isomorphism problem

Sorry if the following are stupid questions (i do not know much about the graph theory).
1. Motivation
we do not know the graph isomorphism problem in class P or NP complete and it is P in the ...

**14**

votes

**3**answers

510 views

### Complexity of equitable partitions

We are talking about undirected simple graphs and partitions of their vertex sets into disjoint non-empty cells. Such a partition is equitable if for any two vertices $v,w$ in the same cell, and any ...

**3**

votes

**1**answer

240 views

### Complexity of Labeled Graph Homomorphism

The following recreational math problem has been floating around work:
We're given an $m \times n$ grid ($m,n$ positive integers). We wish to label the elements of the grid with letters so that we ...

**3**

votes

**1**answer

123 views

### Counting connected fundamental domains of actions on Cayley graphs

The following question arises, for me, from mathematical music theory:
Write $({\Bbb Z}^n,E_n)$ for the Cayley graph of ${\Bbb Z}^n$
relative to standard free generators.
Given a subgroup $L$ of ...

**1**

vote

**1**answer

496 views

### k-uniform k-partite hypergraph matching in polynomial time

I have what seems like an elementary question, but google didn't throw up any answers for it. I would appreciate any pointers that MO users may provide.
It is well known that for $k\geq 3$ finding ...

**3**

votes

**1**answer

125 views

### Quick tests for Self complementary vertex transitive graphs

Are there any quick tests to determine if a graph is Self complementary vertex transitive? That is if the graph is self complementary vertex transitive the answer should be yes.

**3**

votes

**0**answers

154 views

### Best lower bound for proof complexity of graph asymmetry

Graph automorphism problem ( GA) of determining whether a graph has a nontrivial automorphism is a good candidate for a problem in $NP$-intermediate. I'm looking for references that study the ...

**6**

votes

**1**answer

618 views

### NP-hardness of a graph partition problem?

I'm interested in this problem: Given an undirected graph $G(E, V)$, Is there a partition of $G$ into graphs $G_1(E_1, V_1)$ and $G_2(E_2, V_2)$ such that $G_1$ and $G_2$ are isomorphic? Here $E$ is ...

**2**

votes

**2**answers

640 views

### Enumerating all Hamiltonian Cycles in a Bipartite Vertex Transitive Graph

Hi everyone!
This is my first post, apologies if I made any mistakes anywhere.
Here goes the question:
Consider all length 7 binary sequences.
Let $X$ be the set of sequences with hamming weight 3 ...

**13**

votes

**6**answers

1k views

### Compressing Graphs (Kolmogorov complexity of graphs)

What is known about compressing graphs? Here, with "compressing", I mean something like "putting a graph into a zip program"; or with a more technical expression, what is know about the Kolmogorov ...

**2**

votes

**0**answers

180 views

### Complexity of bipartite graphs and their matchings.

My question concerns a hypothetical family of bipartite graphs, $G_i$.
Each graph $G_i$ has $2^i$ red nodes and $2^i$ blue nodes - so nodes get labelled
by their color and a binary string of ...

**4**

votes

**1**answer

826 views

### Finding a vertex of least distance to all other vertices in a graph

Some background to my question: if $G$ is a (simple) graph of $N$ vertices, labelled by integers $0,1,...,N-1$, the closeness centrality of a vertex $i$, denoted by $C(i)$, is defined to be the ...

**2**

votes

**0**answers

339 views

### Hamiltonian paths in subgraphs of rectangular lattice graphs

Is following decision problem NP-hard / NP-complete:
Having vertex-induced subgraph of rectangular lattice graph determine if any Hamiltonian path exists
Having vertex-induced subgraph of ...

**2**

votes

**2**answers

528 views

### Could this be a NP complete?

Given a undirected and unweighted graph G(V,E). M is a subset of vertices of V.
s is a vertex in V - M.
Find an optimal tree T of G defined as:
(1) M and s are in V(T)
(2) Distance (which is ...

**2**

votes

**0**answers

189 views

### Complexity of finding disjoint 2-factors with equal cardinality in cubic graphs?

Finding a connected 2-factor that contains every node in cubic graphs is $NP$-complete since it is equivalent to the Hamiltonian cycle problem. I'm interested in the complexity of finding vertex ...