Questions tagged [graph-isomorphism]

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524 views

How to prove graph isomorphism is NP?

I know that Graph Isomorphism should be able to be verified in polynomial time but I don't really know how to approach the problem. Any help would be appreciated.
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Why the soundness error in the $\mathrm{IP}$ of GNI can implicate $\mathrm{\Sigma_2} \subseteq \mathrm{\Pi_2}$ if GNI is co-NP-Complete?

PDF here shows a way to proof GI is NP-Complete $\implies \Sigma_2 = \Pi_2$. In the last step, it writes following: In other words, (1) is false in this case as required. Book Computational ...
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1answer
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Selecting the right partition in NAUTY

Graph isomorphism solver Nauty has two main procedures, individualization and refinement, to get to a discrete partition. During refinement procedure, we take some cell of the current partition and ...
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1answer
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What is a polynomial-time algorithm for determining whether two trees, with colored nodes, are isomorphic or not

Provide any polynomial-time algorithm (even a large degree polynomial) which determines whether two rooted colored trees are isomorphic to each-other or not. For example, consider the following two ...
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23 views

Directed weighted multigraph isomorphism algorithms

Are there known algorithms for the isomorphism problem for directed weighted multigraphs? If not, could one be created simply by adapting existing algorithms for graphs or digraphs, or is it entirely ...
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Are isomorphic (untyped) lambda expressions semantically equivalent?

"Isomorphic" is defined as having the same shape of syntax trees and the same bindings of variables. However, the variable names might be completely different. In other words, it is to say that we ...
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1answer
152 views

Why is Graph Isomorphism downward self reducible?

To say that graph isomorphism is downward self reducible means the following: There is an algorithm which decided graph isomorhpism for two given graphs of n vertices in polynomial time by accessing ...
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1answer
31 views

Isomorphism problem on bounded degree digraphs

By Babai & Luks (1983), it was proved that graph isomorphism problem is tractable on bounded degree graphs. However, I could not find any result when the graph is edge directed. Is isomorphism ...
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1answer
124 views

Rooted Tree Isomorphism Algorithm

I have developed an algorithm to determine if two rooted trees are isomorphic, which is based on the following conjecture: Let $S_{u}$ be the number of vertices in the rooted subtree of vertex $u$. ...
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2answers
208 views

Isomorphisms between regular graphs of same degree

Are all $n$-vertex regular graphs of degree $d$ isomorphic? Can someone provide an example of two non-isomorphic graphs $G_1$ and $G_2$ which are both regular with degree $d$ and have the same number ...
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What is the current fastest algorithm for finding the maximum common subgraph?

First of all, it's my first time in #ComputerScience at StackExchange so, my apologies if I'm making some newbie mistake when asking this question. So, I'm currently researching algorithms for ...
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1answer
2k views

If graph isomorphism is in P, is then P = NP?

I think that, since graph isomorphism is not known to be $\textbf{NP}$-complete, we can not reduce all problems in $\textbf{NP}$ to it, and therefore the implication does not hold. Additionally, in ...
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65 views

Implementation of Babai's GI algorithm

Babai proved that there is a quasi-polynomial-time algorithm for graph isomorphism. Has anyone implemented this algorithm? I think the code will be easier to understand than his paper, because of the ...
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Complementary of graph invariant

Graph invariant is a property that holds for two isomorphic graphs. For example, degree sequence is graph invariant. We can write $d(G) \ne d(G') \Rightarrow G \ncong G'$, although $d(G) = d(G')$ ...
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3answers
464 views

Hard connected instances for Weisfeiler-Lehman test of isomorphism

There are instances when WL algorithm fails. For example graphs G1 and G2 below have the same coloring after WL-1 algorithm. However, one of these graphs is disconnected. So what are the instances ...
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34 views

Repeated subgraph isomorphism query for single-edge addition for bounded degree graphs

I have a source undirected colored graph $G$ and a base query graph $g$. I know $g$ is subisomorphic to $G$ and now I want to identify which edges I can add to $g$ to preserve subisomorphism. That is, ...
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1answer
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Is distinguishing Hadamard matrices _really_ NP-hard?

In a few different places ( http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01539-4/S0025-5718-03-01539-4.pdf and https://books.google.com/books?id=qYYKBwAAQBAJ&pg=PA21&lpg=PA21&...
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Subgraph Isomorphism checking in Multigraphs

I am considering the following problem: Input: 2 Graphs G=(V,E), H=(V',E'). G and H are directed multigraphs Question: Find a subgraph in G which is isomorphic to H Is there any algorithm ...
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1answer
85 views

What is sub graph isomorphism?

I am confused about the definition of sub graph isomorphism, wikipedia says that the subgraph isomorphism problem is a computational task in which two graphs $G$ and $H$ are given as input, and one ...
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1answer
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Graph isomorphism and the automorphism group

A common approach to decide whether two given graphs are isomorphic is to compute the so-called canonical label (alternatively, canonical graph) of each graph and to check whether those match or not. ...
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48 views

Efficient approximation for find all the nodes and edges which match with some sub-tree in a graph

Let's suppose that I have a big digraph D and a small tree T (small w.r.t D), both directed, D can be connected or not, but T is connected. Here an example: Let's say that D is as follow: And T is ...
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1answer
49 views

Sub-Graph Isomorphism for graphs with multiple edge types and multiple node types

I found that there are algorithms like VFlib and LAD filtering for subgraph isomorphism with one edge type. For multiple node types,one idea could be color all node types with the same type and use ...
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148 views

brute-force canonical labelling of a simple graph

I am looking for an example that shows how to canonically label a graph using brute-force approach. An example with a 3 vertex or 4 vertex simple graph is preferred.
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1answer
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Is it a valid graph canonical form?

This question is motivated from this post. Let $G$ be a given graph, for each vertex $v \in V$, I will label $v$ with $Triangle(v)$. $Triangle(v) : $ means number of distinct triangles contain $v$. ...
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1answer
47 views

Proof that locality is sufficient in showing two graphs are isomorphic

Using the graph representation with (node, [list of neighbours]), to show that two graphs are isomorphic it is sufficient to: show that the vertices have the same degree and for every pair of ...
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1answer
43 views

Find isomorphism of graph with maximal number $x$ such that $f(x)\neq x$ - assuming $NP=P$

For $f : V → V$ which is authomorphism of directed graph $G = (V, E)$, $$\#f = |\{v : f(v) \neq v\}|$$ For graph $G$ we denote: $$\#G = \max\{\#f : \text{$f$ is isomorphism $G$} \}$$ Prove ...
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0answers
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Interactive protocols for showing knots are “hard” to untie

Given two graphs $G_1$ and $G_2$, a zero-knowledge interactive protocol for a prover to convince a verifier that $G_1\not\cong G_2$ entails: The verifier choosing a random $i\in\{1,2\}$ The verifier ...
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1answer
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How similar is the Goldwasser-Sipser Set Lower Bound Protocol to the Hashcash/Bitcoin Proof-of-Work?

Given a hash function $H:\{0,1\}^*\rightarrow\{0,1\}^n$, a difficulty $d\in\mathbb{N}$, and data $D\in\{0,1\}^*$, the framework of the Hashcash/Bitcoin Proof-of-Work entails finding a nonce $c$ such ...
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1answer
48 views

Polynomial-time algorithm for Graph Isomorphism in case of Maximum Constrained Maximum Degree

From Wolfram: A polynomial time algorithm is however known for planar graphs (Hopcroft and Tarjan 1973, Hopcroft and Wong 1974) and when the maximum vertex degree is bounded by a constant (Luks ...
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0answers
110 views

Generating all directed multigraphs

I am trying to find an algorithm that generates all directed multigraphs with a given number of vertices and arcs up to isomorphism (no two generated graphs should be isomorphic). I also want to allow ...
2
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1answer
60 views

Algorithm for getting symetric vertex sets of undirected graph

For my application problem, I am searching for an algorithm that can find all symmetric vertex sets of an undirected labeled graph. My definition of symmetric vertex set is: Let $G$ be a graph with ...
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0answers
104 views

Generate all non-isomorphic bounded-degree rooted graphs of bounded radius

I need to generate/enumerate isomorphism classes of vertex-rooted graphs with the following properties. Let $\Delta$ be the maximal degree (say 3 for subcubic graphs) and $r$ the maximal distance of a ...
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1answer
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Literature about a naive approach to graph isomorphism by inspecting polynomials of adjacency matrices

I describe an approach to graph isomorphism which probably has false positives, and I am curious whether there is literature indicating that it does not work. Given two adjacency matrices $G, H$, an ...
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2answers
6k views

Subgraph isomorphism reduction from the Clique problem

I was trying to understand the Wikipedia proof for NP-completeness of subgraph isomorphism by reduction from the clique problem. It's really just one sentence: Let $H$ be the complete graph $K_k$; ...
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0answers
80 views

Shortest paths in isomorphic graphs with different edge weights

I'm looking for a way to find the shortest paths from a source to all destinations in isomorphic undirected graphs with different edge weights. The only thing I can think of is using Dijkstra on each ...
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1answer
141 views

Find Mapping Node in a Graph

Given two large directed graphs (may have loops and lonely nodes) A and B. They are structurally similar. If we give a node in Graph A, how to find the corresponding one in Graph B? Finally, we need ...
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1answer
136 views

Efficient algorithm for graph canonization for directed acyclic graphs?

I'm interesting in generating directed acyclic graphs (see here, for example). As part of this search, I'm curious if there are any efficient algorithms for determining a canonization of a directed ...
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2answers
467 views

Generating all directed acyclic graphs with constraints

I am interested in listing all the unlabeled1 acyclic digraphs with n vertices which satisfy some additional constraints, such as (a) the resulting graph is connected and (b) except for ...
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1answer
216 views

Automorphism of a Graph with a given Set of Permutations

Given a graph $H$. A set of permutations $\alpha$ which contains permutations of vertices of $H$. The permutation set $\alpha$ has automorphisms of subgraph $H_1, H_2,..... H_x$ where $x$ is the ...
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1answer
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If graph isomorphism yields a polynomial time algorihtm

Greeting I'm studying computing theory and are trying to grasp the concept of complexity classes. If graph isomorphism (suspected NPI) turns out to have polynomial time solution. What possible ...
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1answer
45 views

On graph isomorphism over exponential word sizes

Is it known Graph isomorphism can be done in poly time if we allow exponential word sizes? (Shamir's poly time Integer Factoring algorithm is over exponential word sizes).
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1answer
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Restricting possible permutations in Graph Isomorphism problem

Given a $2n$ vertex undirected graph whose vertices are partitioned arbitrarily in pairs to say WLOG $(1,2)$, $(3,4)$, $\dots$, $(2n-1,2n)$. Call these vertices pairs as super vertices. Call two such ...
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1answer
96 views

On deterministic weighted graph isomorphism from randomized

Is there a $O(n^2)$ algorithm to resolve isomorphism between two weighted $n$-vertex graphs? This is a much easier problem than graph isomorphism. Basically take an real edge weight set $\{w_1,\dots,...
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2answers
408 views

NSPACE for checking if two graphs are isomorphic

I was studying nondeterministic Turing Machines and came across the following question: Describe a nondeterministic Turing Machine (NTM) that only accepts two graphs (G1 and G2) if they are ...
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5answers
897 views

On graph isomorphism for weighted graphs

Is there a $O(n^2)$ algorithm to resolve isomorphism between two weighted $n$-vertex graphs? This is a much easier problem than graph isomorphism. Basically take an real edge weight set $\{w_1,\dots,...
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1answer
1k views

How to build the Reduction from Hamiltonian Cycle problem to Subgraph isomorphism? [duplicate]

I'm trying to prove that the Subgraph isomorphism problem is NPC using the Hamiltonian Cycle problem. Unfortunately I feel (or don't understand) that the solution is "empty" and doesn't explain the ...
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2answers
388 views

counterexample for this graph isomorphism algorithm

I'm trying to learn about graph isomorphism and I stumbled upon coloring. When given 2 graphs, you give each vertex a color according to properties of their neighbors and any vertex on graph 1 can ...
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Isomorphic induced subgraph problem using Courcelle's theorem

The isomorphic induced subgraph problem, is the problem of deciding whether, given two graphs $G$ and $H$, $G$ contains an induced subgraph isomorphic to $H$. Is there a proof using ...
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2answers
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Can the isomorphic graph problem be solved in deterministic polynomial time?

Here is a recent homework problem of mine: Call graphs G and H isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = {⟨G,H⟩| G and H are isomorphic graphs}. Show ...
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0answers
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Implications of Babai's Proof that Graph Isomorphism is Quasi Polynomial Time [closed]

In the context of the very recent talk by Lazlo Babai outlining that Graph Isomorphism (GI) is Quasi Polynomial Time, what are the broader implications of this result? (I'm assuming the claim will ...