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Problem 12

# Call graphs $$G$$ and $$H$$ isomorphic if the nodes of $$G$$ may be reordered so that it is identical to $$H .$$ Let $$I S O=\\{\langle G, H\rangle \mid G$$ and $$H$$ are isomorphic graphs $$\\} .$$ Show that $$I S O \in \mathrm{NP}$$.

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To show that ISO ∈ NP, we provide a non-deterministic polynomial-time algorithm that verifies whether a given pair of graphs is isomorphic or not. The algorithm uses a certificate, which is a permutation vector representing a bijection between the vertex sets of two graphs. It then verifies if the bijection preserves adjacency by checking all pairs of vertices for adjacency in both graphs. The complexity of this algorithm is O(n^2), which is polynomial in both the size of the certificate and the size of the graphs, so ISO ∈ NP.
See the step by step solution

## Step 1: Understand graph isomorphism

Graph isomorphism is a bijection (one-to-one and onto) mapping between the vertex sets of two graphs such that the adjacency between vertices is preserved. In other words, the bijection must map adjacent vertices to adjacent vertices, and non-adjacent vertices to non-adjacent vertices.

## Step 2: Construct a certificate

A certificate for this problem will be a bijection between the vertex sets of two graphs. This can be represented as a permutation vector of the vertices, such that the vertex in the ith position maps to the vertex in the position indicated by the value at the ith index in the permutation vector.

## Step 3: Certificate verification algorithm

Given a graph pair ⟨G, H⟩ and a certificate (permutation vector) for these graphs, we can verify if the graphs are isomorphic by checking if the bijection preserves adjacency as follows: 1. For every pair of adjacent vertices (u, v) in graph G, check if vertices (u', v') are adjacent in graph H, where u' and v' are the vertices in graph H corresponding to u and v, according to the provided permutation vector. 2. For every pair of non-adjacent vertices (u, v) in graph G, check if vertices (u', v') are non-adjacent in graph H, where u' and v' are the vertices in graph H corresponding to u and v, according to the provided permutation vector. 3. If all adjacency checks pass, then the certificate is valid and graphs G and H are isomorphic, otherwise they are not isomorphic.

## Step 4: Analyze the complexity

The certificate verification algorithm takes O(n^2) time, where n is the number of vertices of the graphs, as there are at most n * (n-1) / 2 pairs of vertices to check for adjacency. Since the certificate is of size n and the algorithm is polynomial in both the size of the certificate and the size of the graphs, we conclude that ISO ∈ NP.

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