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Questions tagged [graph-isomorphism]

Two graphs G, H are isomorphic if there is a relabeling of the vertices of G that produces H, and vice-versa. The graph isomorphism problem (GI) is to decide whether two given are isomorphic. In addition to its practical interest, it was identified by Karp in 1972 as having unknown complexity, is one of the few remaining natural candidates for an NP-intermediate problem, and led to the creation of the complexity class AM.

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1 answer
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Is the protocol perfect zero knowledge?

Consider such protocol for $GI$ (Graph-isomorphism problem). $P$ randomly chooses permutations $\sigma_1, \sigma_2, ..., \sigma_k$ and sends $H_1 = \sigma_1(G_0), ..., H_k = \sigma_n(G_0)\ (k > 1)$;...
1 vote
1 answer
177 views

Is (Restricted) Bigraph Isomorphism Weaker than Graph Isomorphism?

I am investigating a paper from Dominik Grezlak and and Uwe Aßmann: “A Canonical String Encoding for Pure Bigraphs.” On page 2, they define the notion of a bigraph, which is roughly a forest and ...
3 votes
0 answers
69 views

Property testing algorithm for isomorphism to a balanced 3-sided complete graph

I am looking for testing algorithm in the dense graph model, that checks for a graph with $3n$ vertices whether it's isomorphic to a balanced 3-sided complete graph with $n$ vertices in each set. The ...
5 votes
1 answer
212 views

Constructing Orbits of the Automorphism of a Graph Group in Bliss

I'm using the Bliss package for graph isomorphism and canonization. The program is working great for the type of graphs I'm interested in. In one of the applications I need to compute the orbits of ...
7 votes
3 answers
336 views

FPT algorithm for Partial k-tree Isomorphism

H.L. Bodlander in Polynomial algorithms for graph isomorphism and chromatic index on partial $k$-trees given a polynomial time algorithm for graph isomorphism when $k$ is constant. Is there any FPT ...
6 votes
0 answers
235 views

Two graphs indistinguishable by 4-WL

There are pairs of non-isomorphic graphs that are indistinguishable by the k-WL (but distinguishable by (k+1)-WL) [1]. For example 4x4 rook’s graph and the Shrikhande graph are non-isomorphic but the ...
0 votes
1 answer
139 views

2xn grid graphs from ring graphs via local complementations

(Local complementation) A local complementation $\tau_v$ is a graph operation specified by a vertex $v$, taking a graph $G$ to $\tau_v(G)$ by replacing the induced subgraph on the neighborhood of $v$, ...
1 vote
2 answers
138 views

On lattice and code isomorphism

We know deciding isomorphism between lattices or codes is difficult if the presentation is through arbitrary bases. What if the presentation of the lattice is through minimum bases? Likewise the ...
39 votes
4 answers
3k views

Techniques for showing that problem is in hardness "limbo"

Given a new problem in $\mathsf{NP}$ whose true complexity is somewhere between $\mathsf{P}$ and being NP-complete, there are two methods that I know of that might be used to prove that resolving this ...
1 vote
0 answers
116 views

Efficient enumeration of connected functional digraphs (up to isomorphism)

Together with the research intern I am supervising, we are currently writing some software that requires us to enumerate all connected functional digraphs of $n$ vertices up to isomorphism (also known ...
13 votes
1 answer
1k views

Is anyone aware of a counter-example to the Dharwadker-Tevet Graph Isomorphism algorithm?

At http://www.dharwadker.org/tevet/isomorphism/, there is a presentation of an algorithm for determining if two graphs are isomorphic. Given a number of shall we say, "interesting" claims by A ...
14 votes
1 answer
355 views

Generating Graphs with Trivial Automorphisms

I'm revising some cryptographic model. To show its inadequacy, I've devised a contrived protocol based on graph isomorphism. It is "commonplace" (yet controversial!) to assume the existence ...
23 votes
2 answers
1k views

Relationship between symmetry and computational intractability?

The $k$-fixed point free automorphism problem asks for a graph automorphism which moves at least $k(n)$ nodes. The problem is $NP$-complete if $k(n)=n^c$ for any $c$>0. However, If $k(n)=O(\log n)$...
17 votes
2 answers
439 views

similar matrices

Given two $n \times n$ matrices $A$ and $B$, the problem of deciding if there exist a permutation matrix $P$ such that $B = P^{-1}AP$ is equivalent to GI(Graph ...
10 votes
2 answers
914 views

Fastest known deterministic algorithm for the undirected Graph Isomorphism problem

What is the fastest known undirected graph isomorphism algorithm?
1 vote
0 answers
68 views

Are there classes where all Eulerian orientations can be listed in polynomial time?

Is there is a subclass of regular graphs (say 4-regular graphs) for which there is a polynomial time algorithm to list all Eulerian orienations? An Eulerian orientaiton of an (undirected simple) graph ...
8 votes
1 answer
497 views

Can you find a counter-example for this proposed Graph Isomorphism algorithm?

As D. Eppstein pointed out here regarding proposed poly-time algorithms for Graph Isomorphism: ... it is easy to define algorithms for graph isomorphism that attempt to amplify some sort of subtle ...
7 votes
1 answer
349 views

Isomorphism of ‘ordered’ DAGs / acyclic semiautomata

I am wondering what is known about the isomorphism problem on ordered DAGs, in particular how to find a canonical representative modulo isomorphism. By ordered I mean that each vertex has a list of ...
0 votes
0 answers
146 views

Comparing two graphs when starting from a single edge

Let's assume that we are given two graphs $G_1$ and $G_2$ defined by the two following nicely drawn pictures. Black numbers label the nodes, red numbers show the edge weight between the nodes. $G_1$ ...
4 votes
1 answer
311 views

Isomorphism preserving transformation CNF to Graph?

In short we are interested in isomorphism preserving transformation CNF to Graph. Let $\phi_1,\phi_2$ be CNF formulas. Define $\phi_1$ and $\phi_2$ to be isomorphic $\phi_1 \cong \phi_2$ if there ...
30 votes
2 answers
895 views

Proof refutation: Amateur reviews of ambitious CoRR papers

I guess that I read too many ambitious CoRR papers. The problem is that those papers are not peer reviewed, but often sound interesting and pass basic plausibility checks. Or maybe they don't, and I ...
7 votes
0 answers
170 views

Subgraph isomorphism on graph sequences

I'm looking for a subgraph isomorphism algorithm that takes advantage of properties of graph sequences. Say $\{G_i\}_{i=1}^k$ is a sequence of graphs on vertex set $\{1 ... n\}$, and two consecutive ...
8 votes
1 answer
276 views

Complexity of graph isomorphism with properly colored edges (ref. request)

An edge-colored graph $G$ is a graph whose edges are labeled with a color (generally represented by an integer). Such a coloring is proper if all adjacent edges in $G$ have different colors. I ...
3 votes
1 answer
141 views

Complexity of unbalanced bipartite isomorphism

For $i=1,2$, let $G_i=(A_i\cup B_i,E_i)$ be an undirected bipartite graph with bipartition $A_i$ and $B_i$, where $|A_1|=|A_2|=a$ and $|B_1|=|B_2|=b$ with $a\le b$. Question. Is the problem of ...
28 votes
4 answers
6k views

What evidence is there that Graph Isomorphism is not in $P$?

Motivated by Fortnow's comment on my post, Evidence that Graph Isomorphism problem is not $NP$-complete, and by the fact that $GI$ is a prime candidate for $NP$-intermediate problem (not $NP$-complete ...
7 votes
1 answer
234 views

Is the isomorphism problem between posets represented by DAGs GI-complete?

Given two directed acyclic graphs, how hard is the problem of checking whether the partial orders they represent are isomorphic? Is this problem GI-complete? I believe this problem is equivalent to ...
10 votes
1 answer
275 views

Efficient graph isomorphism for similar graph queries

Given the graph G1, G2 and G3, we want to perform isomorphism test F between G1 and G2 as well as G1 and G3. If G2 and G3 are very similar such that G3 is formed by deleting one node and inserting one ...
30 votes
1 answer
5k views

What's the status of Babai's Graph isomorphism result?

It's been over a year since his January 2017 retraction and correction. Is there news? If not is this normal for validation to take this long? I would expect it would get plenty of attention. Has ...
1 vote
0 answers
87 views

Graph automorphism with prescribed values

Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
6 votes
1 answer
191 views

Separating words and graph isomorphism

I wonder if there are any known implications of Babai's recent quasi-polynomial time algorithm for Graph Isomorphism to separating words by DFA's. In both cases the ultimate goal is to differentiate ...
1 vote
0 answers
58 views

Is there an efficient way to reduce a set of graphs under isomorphism?

Suppose we have a set $S$ of (fixed-size) graphs, many of which are isomorphic to each other. How do you find a minimum-size set $M$ of graphs such that every $g\in S$ is isomorphic to some graph in $...
3 votes
0 answers
68 views

Pseudodeterministically choosing elements from efficiently samplable distributions (or, the plausibility of a weak choice principle)

Suppose we have a poly-time samplable family of distribution. I.e., a family of distributions $D_n \subseteq \{0, 1\}^{\mathsf{poly}(n)}$ and an algorithm $S$ for which $D_n = (r \leftarrow^\$ \{0,1\}^...
15 votes
2 answers
820 views

GI-hard graph problem not known to be $NP$-complete

Graph Isomorphism ($GI$) is good candidate for $NP$-intermediate problem. $NP$-intermediate problems exist unless $P=NP$. I'm looking for natural problem that is hard for $GI$ under Karp reduction (A ...
3 votes
1 answer
127 views

Isomorphism Problems with Unknown Single-Exponential Algorithms

Isomorphism problems come in several variants: Group isomorphism can be solved in time $n^{O(\log n)}$ Graph isomorphism can be solved in time $n^{\log^{O(1)} n}$ Isomorphism of linear codes can be ...
2 votes
2 answers
256 views

One Generalization of Graph Isomorphism Problem

Say I generalize the language which consists of pairs of isomorphic graphs to take the following form: $GI(f) = \{ (G, H) \mid \exists \psi : V_G \rightarrow V_H, Pr_{a, b \in V_G} ((a, b) \in E_G \...
27 votes
0 answers
1k views

Counting Isomorphism Types of Graphs

Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
12 votes
1 answer
641 views

Complexity of testing if two sets of $m$ points in $\mathbb{R}^n$ differ only by rotation?

Imagine we have two size $m$ sets of points $X,Y\subset \mathbb{R}^n$. What is (time) complexity of testing if they differ only by rotation?: there exists rotation matrix $OO^T=O^TO=I$ such that $X=OY$...
2 votes
0 answers
174 views

Practical polynomial-time implementation of bounded degree graph isomorphishm

There's a well-known article for solving graph isomorphism problem in polynomial time. Many other articles on the subject of isomorphism mention it as a possible "alternative", but note that is not ...
13 votes
2 answers
939 views

Does Babai's quasipolynomial time $\mathsf{GI}$ algorithm actually generate the isomorphism?

I have a (hopefully simple, maybe dumb) question on Babai's landmark paper showing that $\mathsf{GI}$ is quasipolynomial. Babai showed how to produce a certificate that two graphs $G_i=(V_i,E_i)$ for ...
9 votes
1 answer
3k views

Is DAG isomorphism NP-C

Given two directed acyclic graphs $G_1$ and $G_2$, is it NP-Complete to find a one-to-one mapping $f:V(G_1) \rightarrow V(G_2)$ such that $(v_i,v_j) \in D(G_1)$ if and only if $(f(v_i),f(v_j)) \in D(...
-3 votes
1 answer
270 views

On a GI complete class

https://en.wikipedia.org/wiki/Graph_isomorphism_problem#GI-complete_classes_of_graphs says deciding diameter $2$ radius $1$ graph isomorphism is $GI$ complete. Is it possible only the diameter $2$ ...
12 votes
1 answer
278 views

For any two non-isomorphic graphs $G, H$, does there exist a polysize, polylog quantifier depth first order formula which witnesses this?

I want to be very specific. Does anyone know of a disproof or a proof of the following proposition: $\exists p \in \mathbb{Z}[x], n, k, C \in \mathbb{N},$ $\forall G, H \in STRUC[\Sigma_{graph}] (...
4 votes
1 answer
331 views

A Combinatorial algorithm for trivalent graph isomorphism (except some small subclass)

I am currently working on the isomorphism of graphs, hyper-graphs. The graph isomorphism of graphs of degree at most three (trivalent) known to be in $P$. E.M Luks has given an algorithm for trivalent ...
5 votes
0 answers
193 views

Is E.M Luks algorithm for trivalent graph isomorphism parallelizable?

It is still open whether also Luks’ efficient GI algorithm for graphs with bounded degree is parallelizable i.e. NC. I get this from the survey "On Graph Isomorphism for Restricted Graph Classes" by ...
1 vote
0 answers
90 views

Easy instances for coset intersection problem

Coset Intersection Problem Given : $K,H \le S_n$, and $\sigma \in S_n$ Find : $K \cap H\sigma$ Known results are : $n^{O(\sqrt n )}$ time algorithm by L.Babai. $n^{O(1)} m^{O(\sqrt m )}$, where $...
17 votes
1 answer
1k views

How to cite Babai's new graph isomorphism result?

Recently, Babai has published a paper on STOC 2016 claiming that graph isomorphism can be solved in quasipolynomial time. In the beginning of 2017, Babai retracted the quasipolynomial claim due to ...
16 votes
2 answers
3k views

Hard Instances for graph isomorphism testing

Is the case of strongly regular graphs the hardest one for GI testing? where "hardest" is used in some "common sense" meaning, or "in average", so to speak. Wolfram MathWorld mentions some "...
2 votes
1 answer
260 views

Is graph automorphism Karp-reducible to graph isomorphism under hidden subgroup representation?

The classical representations of the graph automorphism problem is Karp-reducible to the classical representation of the graph isomorphism problem. The sketch of proof for this can be written as ...
43 votes
3 answers
3k views

Consequences of a quasi-polynomial time algorithm for the graph isomorphism problem

The Graph Isomorphism problem (GI) is arguably the best known candidate for an NP-intermediate problem. The best known algorithm is sub-exponential algorithm with run-time $2^{O(\sqrt{n \log n})}$. ...
-2 votes
1 answer
269 views

questions on implications Babais quasi P time graph isomorphism result

Babai has reputedly repaired his proof of graph isomorphism in quasipolynomial time.[1] the proof hinges crucially on Johnson graphs. based on the proof, does this mean now that if Johnson graphs can ...