A graph G is isomorphic to a graph H if there is a relabeling of the vertices of G that produces H, and vice-versa. The graph-isomorphism problem (GI) is that of deciding whether graphs G and H are isomorphic.

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Proof of an Ising model representation of graph isomorphism problem

I am going to through Ising formulations of many NP problems by Andrew Lucas. In section $9$ on page 22, the author introduced an exact Ising formulation of the graph isomorphism problem. Given two ...
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270 views

How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is $NPI$ candidate). Restricted GA Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ pairs of nodes $(u, v)$ where $u ...
2
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107 views

When replacing an edge with a graph gadget preserves graph isomorphism?

The transformation replacing an edge by a graph gadget is widely used in graph theory. As an example, in an answer Marzio De Biasi subdivided edges which increases the girth while preserving GI. A ...
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130 views

Does this paper imply graph isomorphism is polynomial for cubic and $4$-regular graphs?

This paper gives example of polynomial GI for certain graphs. Probably I am misunderstanding the paper, but appears to me it implies polynomial GI for cubic, $4$-regular and probably higher degree ...
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“Tiny” Graph Isomorphism

While thinking about the complexity of testing isomorphism of asymmetric graphs (see my related question on cstheory), a complementary question came to my mind. Suppose that we have a polynomial time ...
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53 views

Infinitary Counting Logics: 1-sorted vs. 2-sorted framework

There are two ways to extend infinitary logic with counting: Grädel's way (cf. p. 11): We extend $L_{\infty\omega}$ by introducing a counting existential quantifier: $$ \mathcal{A} \models ...
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130 views

Complexity of counting graph endomorphisms

A homomorphism from a graph $G = (V, E)$ to a graph $G' = (V', E')$ is a mapping $f$ from $V$ to $V'$ such that if $x$ and $y$ are adjacent in $E$ then $f(x)$ and $f(y)$ are adjacent in $E'$. An ...
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109 views

Fixed parameter tractable algorithms for graph isomorphism

What are the future directions in fixed parameter tractability of graphs isomorphism after these two recent papers: Reduction Techniques for Graph Isomorphism in the Context of Width Parameters, ...
12
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187 views

Testing isomorphism of asymmetric graphs

While reading the question Examples where the uniqueness of the solution makes it easier to find, a new (easier?) question came to my mind: actually we don't know if the Graph Isomorphism ($GI$) ...
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158 views

graph isomorphism given a partial isomorphism

Is there an approach to graph isomorphism considering that we are already given a partial isomorphism ? In particular, it would be interesting to have conditions on this partial isomorphism that makes ...
24
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317 views

Can graph isomorphism be decided with square root bounded nondeterminism?

Bounded nondeterminism associates a function $g(n)$ with a class $C$ of languages accepted by resource-bounded deterministic Turing machines, to form a new class $g$-$C$. This class consists of those ...
5
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Probability of random (in)finite graphs being isomorphic

I once skimmed a paper which examined the probability of two (infinite) graphs picked at random being isomorph. The surprising result was that for two random infinite graphs this probability is quite ...
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Chromatic number of a graph where neighbourhood of each vertex is isomorphic

Consider a graph $G(V,E)$. For any two vertices $v_1 , v_2 \in V$, the subgraph induced by the neigbourhood of vertex $v_1$, denoted by $G_{v_1}$ be isomorphic to the subgraph induced by the ...
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53 views

Definition of Clique vertex and separator vertex

While converting a tree decomposition of graph to Normalized tree decomposition, the definitions of clique vertex and separator vertex are used in Sequential and parallel algorithms for embedding ...
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3answers
2k 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 ...
6
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172 views

How hard is the Circuit Isomorphism problem?

Given two circuits, how hard is it to tell if they represent the same function? Clearly, this must be at least as easy as Graph Isomorphism since you can represent any circuit as a graph.
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Open problems related to Graph isomorphism

Presently I am doing literature survey on Graph isomorphism (GI) problem. I would like to know some open questions related to the following What are the graph parameters for which fixed parameter ...
14
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456 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 ...
6
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278 views

Quasi-polynomial time algorithm for graph isomorphism

GI arguably is the best known candidate for NP-intermediate problem. The best known algorithm is sub-exponential algorithm. However, I am not aware of any quasi-polynomial algorithm for it. GI is not ...
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Graph Isomorphism: Polynomial time reduction from GI for disconnected graphs to GI for connected graphs? [closed]

Let the Graph Isomorphism Problem be the problem to decide whether there is a one-to-one mapping between the vertices of two graphs that preserves the edge relations. Let the Graph Isomorphism ...
6
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175 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 ...
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15 views

Number of non-isomorphic codes

We have number of non-isomorphic graphs given in http://planetmath.org/enumeratinggraphs Over an alphabet $q$ how many non-isomorphic $[n,k,d]_q$ codes are possible particularly in the special case ...
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69 views

Fractional chromatic number of Johnson graphs

Johnson graphs have a dual like construction to Kneser graphs in the sense that in Kneser we encode non-intersecting k-sets by joining vertices that represent the sets while in Johnson graphs we ...
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200 views

On the subgraph isomorphism problem

The subgraph isomorphism problem problem is to determine given $G$ and $H$ whether $G$ is a subgraph of $H$. Let $G$ and $H$ be regular graphs with degree of $H$ greater than degree of $G$. Does the ...
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108 views

Cayley subgraph isomorphism and complexity of linear subcode decision

Let $G$ be an undirected Cayley graph over an abelian group. Let $H$ a regular graph whose independence number and chromatic number are known. Let $inj(G,H)$ be the number of injective homomorphisms ...
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378 views

On Graph Isomorphism Complete Problems

I am interested to study Graph Isomorphism (GI) complete problems. In the Paper " Problems Polynomially Equivalent to Graph Isomorphism" by Kellogg S. Booth, (1979), proved that many basic problems ...
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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 ...
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Graph Isomorphism Problem

I am doing some literature review on Graph isomorphism problem. Most of papers which I am reading are written by E.M Luks and Laszlo Babai. These papers uses the high level knowledge of group theory ...
14
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1answer
280 views

Counterexample for Corneil's efficient algorithm for Graph Isomorphism

In the paper An Efficient Algorithm for Graph Isomorphism by Corneil and Gotlieb, 1970 a conjecture was stated upon which the stated algorithm relied for solving GI in polynomial time. Namely: ...
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1answer
290 views

Is graph isomorphism in UP ${\cap}$ coUP?

Is graph isomorphism (the decision problem) in $\mathsf{UP}\cap \mathsf{coUP}$? Here $\mathsf{UP}$ is the class of decision problems accepted by an unambiguous Turing machine (see the complexity zoo). ...
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443 views

History and status of the graph matching problem

Part of the difficulty of finding out more about this problem is that the graph matching problem is different from its much more famous cousin, the matching problem, but hard to be distinguished from ...
5
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1answer
154 views

Canonical labeling of special classes of DAGs

Graph Isomorphism of directed acyclic graphs (DAGs) is known to be GI-complete. So a polynomial time algorithm to canonize DAGs is not known. What are some special classes of DAGs that can be ...
4
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1answer
257 views

How hard is to compute $\Delta_{|V|}$?

Let $G=(V,E)$ be a graph. Let $\Delta_k$ be the quantity defined in this question. Let $\mathcal{C}$ be the set of vertex covers of $G$. The following holds: $$ |\mathcal{C}| = 2^{|V|} - \sum_{k = ...
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Equivalence relations on strongly regular graphs with same parameters

Let $\mathcal{S}$ be the set of all strongly regular graphs with parameter $(n, k, \lambda, \mu)$. Are there any (interesting) equivalence relations defined on this set? My motivation is to approach ...
4
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215 views

Optimal upper bound on the number of non-isomorphic graphs with certain parameter

What are the optimal (or best known) bounds (preferably exact or else asymptotic but not expectation on random graphs) on the number of non-isomorphic (unlabelled) simple (no self-loop), undirected ...
10
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297 views

automorphism in Cai-Furer-Immerman gadgets

In the famous counter example for graph isomorphism via Weisfeiler-Lehman (WL) method the following gadget was constructed in this paper by Cai, Furer and Immerman. They construct a graph $X_k = (V_k, ...
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Negative results on identical particles approach to Graph Isomorphism (GI) problem

There has been some efforts to attack graph isomorphism problem using quantum random walk of hard-core bosons (symmetric but no double occupancy). Symmetric power of adjacency matrix, which seemed ...
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Is there any special property the resulting graph G' has?

Undirected graph $G$ can be partitioned into several vertex blocks, each vertex pair $(u,v)$ has an edge if "$u$" and "$v$" are in the different blocks; no edge, otherwise. That is, each block pair ...
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329 views

Approaches to GI inspired by knot problem

GI and Knot Problem both are problem of deciding structural equivalence of mathematical objects. Are there any results establishing connections between them? Nice connections of knot problem to ...
11
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1answer
303 views

Redundancy and Structure of computational problems

It is widely believed that some computational problems such as graph isomorphism can not be NP-complete because it does not possess enough structure or redundancy to be computationally hard (NP-hard). ...
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Strongly Regular Graph and GI-Completeness

It is not known if graph isomorphism (GI) for strongly regular graphs (SRGs) is in P. Are there any hints that it might or might not be GI-Complete? Are there any strong consequences in such cases? ...
10
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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 ...
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Complexity of algorithm to test if a graph is asymmetric

Counting the order of automorphism group of a graph is polynomial-time equivalent to graph isomorphism problem. But if we just want to know if the order is greater than 1, what is the complexity of ...
11
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1answer
512 views

Gentle introduction to graph isomorphism for bounded valance graphs

I am reading about classes of graphs for which Graph Isomorphism ($GI$) is in $P$. One of such case is graphs of bounded valence (maximum over degree of each vertex) as explained here. But I found it ...
15
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373 views

Imperfect subgraph isomorphism

Consider the following problem: Given a query graph $G = (V, E)$ and a reference graph $G' = (V', E')$, we want to find the injective mapping $f : V \rightarrow V'$ which minimizes the number of edges ...
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184 views

Complexity of finding if a degree bounded graph H is a subgraph of an unbounded graph G

You are given two graphs G and H , and want to know if H is a subgraph of G. You know that H has a max vertex degree K (constant integer). What can you say about the complexity of this? I know that ...
14
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1answer
484 views

Hardness of Computing Weisfeiler-Lehman labels

The 1-dim Weisfeiler-Lehman algorithm (WL) is commonly known as canonical labeling or color refinement algorithm. It works as follows : The initial coloring $C_0$ is uniform, $C_0(v) = 1$ for all ...
15
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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 ...
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Is there a gap amplification type of result for the Graph Isomorphism Problem?

Suppose $G_1$ and $G_2$ are two undirected graphs on vertex set $\{1, \dotsc, n\}$. The graphs are isomorphic if and only if there is a permutation $\Pi$ such that $G_1 = \Pi(G_2)$, or more formally, ...
6
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1answer
244 views

Best lower bound for proof complexity of graph non-automorphism problem

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 ...