  This paper is the first from serial papers that provide constructive
characterizations for classes of bidirected graphs known as radials and
semiradials. In this paper, we provide constructive characterizations for five
principle classes of radials and semiradials to be used for characterizing
general radials and semiradials. A bidirected graph is a graph in which each
end of each edge has a sign $+$ or $-$. Bidirected graphs are a common
generalization of digraphs and signed graphs. We define a new concept of
radials as a generalization of a classical concept in matching theory, critical
graphs. Radials are also a generalization of a class of digraphs known as
flowgraphs. We also define semiradials, which are a relaxed concept of radials.
We further define special classes of radials and semiradials, that is, absolute
semiradials, strong and almost strong radials, linear semiradials, and
sublinear radials. We provide constructive characterizations for these five
classes of bidirected graphs. Our serial papers are a part of a series of works
that establish the strong component decomposition for bidirected graphs.
