  Exact pattern matching in labeled graphs is the problem of searching paths of
a graph $G=(V,E)$ that spell the same string as the pattern $P[1..m]$. This
basic problem can be found at the heart of more complex operations on variation
graphs in computational biology, of query operations in graph databases, and of
analysis operations in heterogeneous networks, where the nodes of some paths
must match a sequence of labels or types. We describe a simple conditional
lower bound that, for any constant $\epsilon>0$, an $O(|E|^{1 - \epsilon} \,
m)$-time or an $O(|E| \, m^{1 - \epsilon})$-time algorithm for exact pattern
matching on graphs, with node labels and patterns drawn from a binary alphabet,
cannot be achieved unless the Strong Exponential Time Hypothesis (SETH) is
false. The result holds even if restricted to undirected graphs of maximum
degree three or directed acyclic graphs of maximum sum of indegree and
outdegree three. Although a conditional lower bound of this kind can be somehow
derived from previous results (Backurs and Indyk, FOCS'16), we give a direct
reduction from SETH for dissemination purposes, as the result might interest
researchers from several areas, such as computational biology, graph database,
and graph mining, as mentioned before. Indeed, as approximate pattern matching
on graphs can be solved in $O(|E|\,m)$ time, exact and approximate matching are
thus equally hard (quadratic time) on graphs under the SETH assumption. In
comparison, the same problems restricted to strings have linear time vs
quadratic time solutions, respectively, where the latter ones have a matching
SETH lower bound on computing the edit distance of two strings (Backurs and
Indyk, STOC'15).
