  We introduce a new class of matroids, called graph curve matroids. A graph
curve matroid is associated to a graph and defined on the vertices of the graph
as a ground set. We prove that these matroids provide a combinatorial
description of hyperplane sections of degenerate canonical curves in algebraic
geometry. Our focus lies on graphs that are 2-connected and trivalent, which
define identically self-dual graph curve matroids, but we also develop
generalizations. Finally, we provide an algorithm to compute the graph curve
matroid associated to a given graph, as well as an implementation and data of
examples that can be used in Macaulay2.
