Questions tagged [isomorphism-testing]

Algorithmic questions concerning isomorphism testing. A prime example is the graph isomorphism problem, which is to decide if two input graphs are isomorphic. This tag may also be used for isomorphism testing between other objects (such as groups).

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150 views

Categorical setting for cancellation in direct sums

I am wondering whether some criterion can be put on a category $\mathcal{C}$ with direct sums to ensure that for three objects $X,Y,Z$ one has $$ X\oplus Y \cong X \oplus Z\Longrightarrow Y\cong Z. $$ ...
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51 views

Reduction graph isomorphism to maximum independent set in very dense graph

We got a reduction graph isomorphism to MIS in a very dense graph, or alternatively negative monotone 2-CNF to MAX-ONEs with a formula with many clauses. Let $G,H$ be graphs of order $n$ and adjacency ...
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83 views

Complete graph invariant based on integer programming?

Roughly speaking, we are trying to find complete graph invariant as the lexicographically first matrix from the permutations of the adjacency matrix. Let $G$ be graph, possibly directed graph, of ...
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1answer
737 views

Finding isomorphism between $\mathbb{Z}^2\ltimes_{A,B} \mathbb{Z}^4$ and $\mathbb{Z}^2\ltimes_{A,C} \mathbb{Z}^4$

Let $A,B(a,b,c,d)\in\mathsf{GL}(4,\mathbb{Z})$ be given by $$A=\begin{pmatrix} I_2 & \begin{pmatrix} 0&0\\0&1 \end{pmatrix} \\0& -I_2\end{pmatrix},\quad B(a,b,c,d)=\begin{pmatrix} -2a-...
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What it is wrong with the proof $PH = AM = coAM$ based on FORMULA ISOMORPHISM?

Confusion is possible, but Emil's comment and a paper imply major result: collapse of the polynomial hierarchy. Q1 What is wrong with the proof that $PH = AM = coAM$ given below? In a comment about ...
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210 views

Can we make cryptography signature algorithm based on hardness of isomorphism?

In public key cryptography, Alice knows functions $f$ and its inverse $f^{-1}$. $f$ is public and $f^{-1}$ is secret. To sign a message $m$, she gives $(m,a=f^{-1}(m))$. To verify a signature, the ...
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1answer
124 views

Doubt about lemma for polynomial equivalence [closed]

Multivariate polynomials $f,g$ are equivalent if there exists invertible linear transformation $A$ such that $f(X)=g(A\cdot X)$ From paper p.1: Lemma 1.1. (Structure of quadratic polynomials). Let $F$...
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2answers
465 views

A “subtle” isomorphism testing problem: $\mathbb{Z}\ltimes_{A} \mathbb{Z}^5\cong \mathbb{Z}\ltimes_{B}\mathbb{Z}^5$ or not?

EDIT: I've made a mistake with the matrices. Now it is corrected. A couple of days ago I asked this question. There, answerers gave me excellent hints to solve that case and others too. But I've found ...
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61 views

Isomorphism preserving transformation graph to graph of logarithmic boolean width and bounded degeneracy

The paper On graph classes with logarithmic boolean-width claims that some graph problems are fixed parameter tractable with parameter the boolean width. In particular, boolean-width of the complement ...
3
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1answer
107 views

Weisfeiler-Lehman test for hypergraphs

The Weisfeiler-Lehman test for graph isomorphism is based on iterative graph recoloring and works for almost all graphs, in the probabilistic sense. If we extend the domain to general hypergraphs, ...
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1answer
118 views

Define a homomorphism of a set of graphs to its power set

Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is, $G_1\cup G_2$ $=\langle V(G_1)\cup V(G_2), (E(G_1)\...
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64 views

Polynomial Graph Isomorphism from Star System Reconstruction?

Confusion is possible, but two papers and a simple graph transformation imply Graph Isomorphism is polynomial, which is an open problem. The closed neighborhood of a vertex in a graph is sometimes ...
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115 views

Is bounded graph isomorphism $NP$ complete?

Fix a matrix $M\in(\mathbb Z\backslash\{0\})^{n\times n}$ where $\|M\|_\infty\leq 2^{poly(n)}$. Is the bounded graph isomorphism problem Given symmetric $A,B\in\{0,1\}^{n\times n}$ and $U,V>0$ ...
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451 views

Algorithms for rooted directed acyclic graph isomorphism

Given two directed acyclic graphs $G_1$ and $G_2$, and their roots $r_1$ and $r_2$, is there a polynomial algorithm to determine if $G_1$ and $G_2$ are isomorphic?
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1answer
73 views

Canonical form for a bipartite graph

I have a bipartite graph, including V1 and V2 vertices, and I would like to convert it to a canonical form. One simple method is converting this graph to a general graph by expanding its adjacency ...
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269 views

Canonical labeling of graphs

Definition : Let $V$ be a linearly ordered set and $\Sigma$ an alphabet. A $\Sigma$- string on $V$ is a function $x : V \mapsto \Sigma$. The set of all $\Sigma$- strings on $V$ is defined by $\Sigma^{...
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1answer
122 views

Doubt on size of the automorphism of regular graphs [closed]

Let $X= (V,E)$ be a input graph with vertex set $V$ and edge set $E$. Now we can define the AUT$(X)= \{\sigma | X^{\sigma} = X\}$. We know that if $X= K_n$ (complete graph with $n$ vertices) its AUT$...
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1answer
212 views

Some doubts on the algorithm of graph isomorphism of bounded degree graph

There is an well known algorithm given by E.M Luks for bounded graph isomorphism. There are three important steps of the algorithm two of them are given below: If the group action on vertex set of ...
3
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1answer
163 views

How to find a good individualising set in graph isomorphism

The procedure called individualization breaks symmetry arbitrarily. It chooses some nodes in the graph, arbitrarily, to give their own unique names. Suppose there exists a set $S \subset V$, such that ...
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1answer
115 views

Lower bound on the individualization set for $k$-iso-regular graphs ( degree at max three )?

Assume graphs of degree at most three for this question. A graph is said to be k-isoregular if for every subset $S$ of at most $k$ vertices the number common neighbors of the elements of $S$ depends ...
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3answers
1k views

Are regular graphs the hardest instance for graph isomorphism?

Regular graphs are the graphs in which the degree of each vertex is the same. The Weisfeiler-Lehman algorithm fails to distinguish between the given two non-isomorphic regular graphs. Is there a ...
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1answer
488 views

Elliptic Curve, characteristic equation of Frobenius endomorphism relation to isogeny

Let E be an elliptic curve over $F_p$. Suppose that its j invarient is not supersingular and that $j\neq 0 $ or 1728. Then the modular polynomial $\Phi_l(j,T)$ has a zero $\tilde{\jmath} \in \...
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1answer
70 views

Number of non-isomorphic block graphs on n nodes

A block in a graph is a maximal connected subgraph that has no cut-vertex. A complete graph having $n$ nodes is denoted by $K_n$. A block graph is a graph in which each block is a complete graph. For ...
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1answer
75 views

Complexity of quadratic polynomials isomorphism

Two polynomials $f,g$ are isomorphic iff $f(x_1,\ldots x_n)=g(\pi(x_1, \ldots x_n))$ for a permutation $\pi$. $f,g$ are equivalent if there exists invertible linear transormation $A$ such that $f(X)=...
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2answers
328 views

How to show the two convex bodies are affinely isomorphic?

This problem comes from the response of the author of papers. Consider two convex bodies $A$ and $B$: $$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$ $$B = \operatorname{...
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3answers
549 views

if two graphs have the same distinct eigenvalues, can we conclude that they are isomorphic?

i think i saw something like the statement of the question above but i am not sure. Given two graph G and H if they have the same characteristic polynomial and it does not have any repeated roots,...
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1answer
120 views

a space isomorphic to $S^{p+q}$

I've asked this question few days ago in MathStackExchange, but there was no result. So, I decided to ask it there. In one of the paper I have met that $$\mathbb{S}^{p+q} \cong \mathbb{S}^p \times \...
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0answers
108 views

Isomorphism with fixed number of Permutations [closed]

Suppose, I have a fixed number of permutations for each sub-graph to determine isomorphism of whole graph. Is it possible to determine efficiently ? For example, $G, H$ are isomorphic graphs. For ...
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223 views

Is there an efficient algorithm for testing isomorphism of projective planes?

Isomorphism testing is a core problem in computational complexity. Recently, Babai has shown that Graph Isomorphism problem for general graphs can be solved in quasipolynomial time. Long time before ...
11
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3answers
410 views

Finite objects for which isomorphism is NP-hard or harder?

Are there finite objects for which deciding isomorphism is NP-hard or harder? Graphs and groups are not solutions. Searching the web didn't return answer for me. Partial result based on Chow's ...
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1answer
120 views

Counting bounded genus non-isomorphic graphs

What is the number of non-isomorphic $2n$ vertex balanced bipartite graphs of degree at most $d$ and genus $g$? I am most interested in $d\leq3$ and $g=0$.
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2answers
254 views

Practical permutation search problems resilient to backtrack techniques

I asked a similar question here: Permutation search problems with no known $o(n!)$ algorithms; to which one-way functions over a space of permutations gives a problem with no $o(n!)$ solution (...
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1answer
238 views

Graph Isomorphism for Triangle Free graph

Is there any specific computational complexity result of Graph Isomorphism for Triangle Free graphs? Anything close to the subject will help and of course, I have searched Google.
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2answers
261 views

Permutation search problems with no known $o(n!)$ algorithms

I am looking for problems for whose solution no known subfactorial algorithms are known. I am particularly interested in questions of isomorphism; that is, is there a permutation that converts one ...
3
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1answer
114 views

A possible GI isomorphic problem

Here I try to seek if restricting the structure of permutations would still keep GI property. Given a $2n$ vertex undirected graph whose vertices are partitioned arbitrarily in pairs to say WLOG $(1,...
6
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1answer
245 views

Deciding isomorphism between graphs which interpret in the pure set

I am interested in the following decision problem: Given descriptions of two graphs $G,H$ which interpret in the pure set $\mathbb N=(\{0,1,2,\ldots\},=)$, decide whether $G$ and $H$ are isomorphic....
5
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1answer
139 views

Characterization of non-isomorphic graphs but isomorphic total graphs?

Given a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$. ...
6
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0answers
104 views

Contradicting claims about complexity of directed path graphs isomorphism

Thesis and a paper give conflicting claims about the complexity of graph isomorphism for directed path graphs. Since this means GI is polynomial likely I am missing something or there is something ...
2
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1answer
570 views

Why is graph automorphism sometimes easier than canonical labeling (for current software)?

László Babai recently hinted that graph isomorphism is solved for all practical purposes: It seems, for all practical purposes, the Graph Isomorphism problem is solved; a suite of remarkably ...
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1answer
1k views

Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?

In http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/ it is mentioned: 'In discussing Johnson graphs, Babai said they were a source of “unspeakable ...
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0answers
87 views

What is wrong with this isomorphism preserving transformation to a graph of bounded clique width and bounded rank width?

Got an isomorphism preserving transformation to a graph of bounded clique width and rank width. It, a paper and graphclasses.org imply graph isomorphism is in P, so likely something is wrong. Let $G$ ...
5
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3answers
275 views

Repository of graph classes that are tough to test non-isomorphic pairs from isomorphic pairs

(1) Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs? (2) Is there a repository of adjacencies from such classes?
5
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2answers
250 views

Group acting on two different sets, can isomorphisms be computed efficiently?

For the permutation group isomorphism problem, a permutational isomorphism between two permutation groups is sought. A closely related but seemingly easier problem arises, if an isomorphism between ...
4
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0answers
292 views

Edit distance vs. canonical adjacency matrix distance

Let $G$ and $G'$ be two simple random graphs on the same set of nodes. Let $d_{edit}$ be the edit distance between $G$ and $G'$. Let $\mathbf{A}$ and $\mathbf{A'}$ be the adjacency matrices of the ...
3
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3answers
562 views

Algorithm to determine isomorphism of 2 maximal planar graphs

I read on wikipedia that there are efficient algorithms to answer the question whether 2 (maximal) planar graphs F and G are isomorphic. However, after some (IMHO) substantial searching I don't seem ...
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0answers
226 views

“Graph Individualization”[ reference request] [closed]

Context: I am studying Weisfeiler Lehman method(WL method) and have clear idea about 1 and 2 dimensional WL method. I was wondering about the individualization process described below- ...
9
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1answer
401 views

Relationship of Weisfeiler-Lehman algorithm to weak isomorphism of coherent algebras

A coherent algebra is a matrix algebra (over $\mathbb{C}$) closed under conjugate transpose and Schur (entrywise) product, and that contains the identity matrix $I$ and all ones matrix $J$. Given ...
2
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0answers
440 views

Partitioning graph for Graph Isomorphism [closed]

Motivation: I am studying the graph isomorphism problem. I am trying to construct a partitioning method to reduce search cases . Construction: $G$ is an $r$ regular graph, $k$ connected (not a ...
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0answers
165 views

Is there a polynomial time algorithm for Poly-trees (Oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
8
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2answers
228 views

Isomorphism problem on the class of induced subgraphs of a hypercube

A problem that I am currently studying translates to the problem of deciding whether two induced subgraphs of the hypercube $Q_k$ are isomorphic. Now it feels to me that this class of graphs is "too ...