# Questions tagged [graph-isomorphism]

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### Enumerate all non-isomorphic graphs of a certain size

I'd like to enumerate all undirected graphs of size $n$, but I only need one instance of each isomorphism class. In other words, I want to enumerate all non-isomorphic (undirected) graphs on $n$ ...
4answers
2k views

### Has the graph isomorphism problem been solved?

Wikipedia's graph isomorphism problem page would seem to indicate that, no, it has not been solved. However, a friend of mine pointed out A Polynomial Time Algorithm for Graph Isomorphism . I am not ...
2answers
768 views

### Group isomorphism to graph ismorphism

In reading some blogs about computational complexity (for example here)I assimilated the notion that deciding if two groups are isomorphic is easier than testing two graphs for isomorphism. For ...
2answers
3k views

### Graph isomorphism problem for labeled graphs

In the case of unlabeled graphs, the graph isomorphism problem can be tackled by a number of algorithms which perform very well in practice. That is, although the worst case running time is ...
2answers
1k views

### How to define a similarity between two graphs?

Let's say we have a set of vertices $V$, and two (undirected) graphs over the same set $V$, but not necessarily the same set of edges $G_1 = (V, E_1)$, $G_2 = (V, E_2)$. $\newcommand\mG{\mathbb G}$(...
1answer
144 views

### 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 ...
2answers
283 views

### How hard is it to solve for $P$ in $A = PBP^{-1}$?

From graph isomorphism, we know that two graphs A and B are isomorphic if there is a permutation matrix P such that $A = P \times B \times P^{-1}$ So, to solve the problem, if two graphs are ...
1answer
240 views

### 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. ...
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 ...
1answer
107 views

### 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&...
1answer
243 views

### Subgraph Isomophism Problem - Color Coding Technique - Proof Sketch

I am reading the paper Color Coding by Alon, Yuster, and Zwick. They state a theorem (6.3) that says if $H$ is a graph on $k$ vertices with treewidth $t$ and $G = (V, E)$, then a subgraph of $G$ ...
1answer
1k views

### Subgraph isomorphism in planar graphs

I'm a computer engineer trying to understand this Eppstein paper for matching subgraphs in planar graphs. I'm trying to find subgraph matches to map an application graph (the subgraph) to a network-...
1answer
477 views

### 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 ...
1answer
52 views

### Is complexity of $GI_{di}$ same as $GI_{un}$?

Does the graph isomorphism problem for directed graphs($GI_{di}$) reduce to the graph isomorphism problem for directed graphs($GI_{un}$)? It is clear $$GI_{un}\leq GI_{di}$$ since the set of ...
2answers
185 views

### Complete graph invariant

Does anybody know whether the multiset of the determinants (possibly together with the order of the submatrix they refer to) of all the principal minors of the (symmetric) adjacency matrix of a graph ...
1answer
79 views

### Subgraph isomorphisms: does large out-expansion imply large in-expansion?

Let $G$ be a directed graph, and $H$ a subgraph of $G$ that contains all the vertices of $G$. (In other words, $H$ is obtained by deleting some of the edges of $G$, but not any of the vertices of $G$.)...
0answers
177 views

### 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 ...
2answers
3k views

### 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 ...
2answers
9k 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 $K_k$; ...
2answers
494 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 ...
1answer
382 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 ...
0answers
138 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 ...
0answers
96 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 ...
3answers
3k views

### Finding an isomorphism between finite automata

Im having trouble figuring out how to determine if two finite automata are the same apart from renumbered states. More specifically, heres an example: It's easy to generate a regular expression ...
2answers
701 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 ...
1answer
229 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 $u$. ...
1answer
169 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 ...
1answer
203 views

### Common subgraph isomorphism with K vertex

I'm looking for subgraph isomorphism of at least K vertex between Graph A and B. I only can come up with the dumbest algorithm, which is: Compute all combination of vertices with length K of Graph A. ...
1answer
128 views

### 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 ...
2answers
401 views

### 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 ...
0answers
26 views

### 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 $d(G) = d(G')$ ...
0answers
51 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 ...
0answers
270 views

### Enumerate all non-isomorphic graphs of size n

I am trying to enumerate all non-isomorphic graphs of size n and found this question: Enumerate all non-isomorphic graphs of a certain size The accepted answer ...
3answers
762 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 ...
2answers
459 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 ...
2answers
897 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 ...
1answer
64 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 ...
1answer
54 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).
1answer
45 views

### 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 ...
1answer
51 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 ...
1answer
53 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 ...
1answer
227 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 ...
0answers
23 views

### Subgraph isomorphism index/precomputation

I'm currently working on problem in which a set of graphs $T=\{t_1,\dots,t_n\}$ is given and fixed. Given a graph $m$ I want to check which of the $t_i$ are subgraphs of it, as quick as possible. Is ...
0answers
50 views