I recently read in the paper "Quantum NP is hard for PH" by S. Fenner et. al. that "graph non-isomorphism is known to be in coC_=P", but they did not attach a reference. I have tried looking and I cannot find why this is true. Can anyone help me find out why this is true?
2 Answers
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I don’t know if this is what they had in mind, but the following should work.
By attaching a vertex connected to everybody, ensure that the input graphs $G_0$ and $G_1$ are connected. Let $\def\S#1{\lvert#1\rvert}\DeclareMathOperator\aut{Aut}\aut(G)$ denote the automorphism group of $G$. Then $$\S{\aut(G_0\mathbin{\dot\cup}G_1)}=\begin{cases} \phantom2\S{\aut(G_0)}\,\S{\aut(G_1)}&\text{if $G_0\not\simeq G_1$,}\\ 2\S{\aut(G_0)}\,\S{\aut(G_1)}&\text{if $G_0\simeq G_1$.} \end{cases}$$ Thus, you can test graph isomorphism using the $\mathsf{C_=P}$ predicate $$\S{\aut(G_0\mathbin{\dot\cup}G_1)}=\S{\{0,1\}\times\aut(G_0)\times\aut(G_1)}.$$ Both graph isomorphism and its complement are in $\mathsf{C_=P}$ by this argument.
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1$\begingroup$ Is it clear that attaching a vertex will preserve non-isomorphism? $\endgroup$Wojowu– Wojowu2025-12-18 10:48:14 +00:00Commented yesterday
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3$\begingroup$ Yes. Any isomorphism must map this vertex to a vertex that is connected to all other vertices in the other graph; if this happens not to be the new vertex that was added, you can swap the two by an automorphism to make it so. $\endgroup$Emil Jeřábek– Emil Jeřábek2025-12-18 10:52:57 +00:00Commented yesterday
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1$\begingroup$ That graph isomorphism reduces to isomorphism of connected graphs is in any case well known. Also, the counting argument I’m using here is really the same as the approximate counting argument used to prove that GI is in coAM. $\endgroup$Emil Jeřábek– Emil Jeřábek2025-12-18 11:06:19 +00:00Commented yesterday
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When looking for this paper, I found what seems to be a revised version of it on arXiv, titled Determining Acceptance Possibility for a Quantum Computation is Hard for the Polynomial Hierarchy. In that paper, when mentioning the graph nonisomorphism problem they reference the book "The Graph Isomorphism Problem: Its Structural Complexity". Corollary 4.34 in that book is exactly the statement you ask for. The original reference the book provides is the paper Graph isomorphism is low for PP, which shows that the graph isomorphism problem is not only in $\mathrm{C_=P}$, but is in fact low for it.
Added: their results are actually much stronger than just membership in $\mathrm{C_=P}$, and in particular the algorithms they provide have the gap between accepting and rejecting computations be either $0$ or a value depending only on the sizes of graphs. I believe this is necessary for their lowness results, but if you just want membership in $\mathrm{C_=P}$, then Emil's answer is much simpler.
