In a variety of emerging applications one needs to decide whether a graph
G matches
another
G
p
,
i.e.
, whether
G
has a topological structure similar to that of
G
p
. The traditional notions of graph homomorphism and isomorphism often fall short of capturing the structural similarity in these applications. This paper studies revisions of these notions, providing a full treatment from complexity to algorithms. (1) We propose
p-homomorphism (p
-hom) and 1-1
p
-hom, which extend graph homomorphism and subgraph isomorphism, respectively, by mapping
edges
from one graph to
paths
in another, and by measuring
the similarity of nodes
. (2) We introduce metrics to measure graph similarity, and several optimization problems for
p
-hom and 1-1
p
-hom. (3) We show that the decision problems for
p
-hom and 1-1
p
-hom are NP-complete even for DAGs, and that the optimization problems are approximation-hard. (4) Nevertheless, we provide approximation algorithms with
provable guarantees
on match quality. We experimentally verify the effectiveness of the revised notions and the efficiency of our algorithms in Web site matching, using real-life and synthetic data.
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