  Graph matching aims to find the latent vertex correspondence between two
edge-correlated graphs and has found numerous applications across different
fields. In this paper, we study a seeded graph matching problem, which assumes
that a set of seeds, i.e., pre-mapped vertex-pairs, is given in advance. While
most previous work requires all seeds to be correct, we focus on the setting
where the seeds are partially correct. Specifically, consider two correlated
graphs whose edges are sampled independently from a parent \ER graph
$\mathcal{G}(n,p)$. A mapping between the vertices of the two graphs is
provided as seeds, of which an unknown $\beta$ fraction is correct. We first
analyze a simple algorithm that matches vertices based on the number of common
seeds in the $1$-hop neighborhoods, and then further propose a new algorithm
that uses seeds in the $2$-hop neighborhoods. We establish non-asymptotic
performance guarantees of perfect matching for both $1$-hop and $2$-hop
algorithms, showing that our new $2$-hop algorithm requires substantially fewer
correct seeds than the $1$-hop algorithm when graphs are sparse. Moreover, by
combining our new performance guarantees for the $1$-hop and $2$-hop
algorithms, we attain the best-known results (in terms of the required fraction
of correct seeds) across the entire range of graph sparsity and significantly
improve the previous results in
\cite{10.14778/2794367.2794371,lubars2018correcting} when $p\ge n^{-5/6}$. For
instance, when $p$ is a constant or $p=n^{-3/4}$, we show that only
$\Omega(\sqrt{n\log n})$ correct seeds suffice for perfect matching, while the
previously best-known results demand $\Omega(n)$ and $\Omega(n^{3/4}\log n)$
correct seeds, respectively. Numerical experiments corroborate our theoretical
findings, demonstrating the superiority of our $2$-hop algorithm on a variety
of synthetic and real graphs.
