# Questions tagged [graph-isomorphism]

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### Efficiently generating the smallest directed graph subject to degree constraints that yields a requested flow

Given two (small-ish) sequences $I$ and $O$ of rational numbers with $\sum_{x \in I}x=\sum_{y \in O}y$, generate a directed graph $G = (V,E)$ with minimum $|V|$ that respects the following: $|I|$ ...
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1 vote
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### Weisfeiler-Leman Algorithm

We know that the Weisfeiler-Leman Algorithm will not always distinguish graphs that are not isomorphic but if two graphs are isomorphic are we guaranteed to get a certificate?
1 vote
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### GNI public coin interactive proof: why randomize y?

I've read this scribe that provides a public coin interactive proof for graph non-isomorphism. In the proof, the verifier samples both a pairwise-independent hash function and a target $y$. Then it ...
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### How does the standard proof that IP is in PSPACE apply to, say, the graph non-isomorphism problem?

I learned years ago that $IP \subseteq PSPACE$ since you could simulate all sets of messages and then determine the probability of success. But lately I’ve been looking at the standard proof of this ...
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### Is graph isomorphism $P$-hard?

Intuitively speaking, it would seem like the graph isomorphism problem (which might be $NP$-intermediate) should be $P$-hard. But maybe it's not? Or maybe it's an open question? If it is indeed $P$-...
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### Largest isomorphic subgraphs of two graphs with features

the following question came up in a problem I am working on: Suppose you have two graphs $G_1=(V_1, E_1), G_2=(V_2,E_2)$ that have features attached to them, i.e. to every $v\in V_1$ or $v\in V_2$ ...
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### Automata: Are there algorithms to judge whether two automata are isomorphic?

When I want to judge whether two regular forms represent the same language, I have learned the next method: create the (non-deterministic) finite-state automata which accepts the language the given ...
• 1
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### Subgraph Isomorphism Problem NP complete?

I found many solution online on how to reduce Subgraph Isomorphism problem to Clique, but how do I prove that it is NP complete by reduction from independent set? I'm struggling to figure out this ...
1 vote
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### Constructive Graph Isomorphism Algorithm Literature

I have consulted the literature concerning graph isomorphism algorithms, and all papers I could find involve finding a canonical representation of a graph. So to decide whether two graphs are ...
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• 395
1 vote
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### 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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### 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 ...
• 351
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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 ...
1 vote
47 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
• 351
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### 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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### 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 available ...
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### 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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### 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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### 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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1 vote
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### 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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### 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.
1 vote
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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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1 vote
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### 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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### 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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1 vote
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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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### 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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### 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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### 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 ...
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### 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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### 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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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 ...