A theta graph 
is a graph formed by joining two distinct vertices
by three internally disjoint paths of lengths 
, 
, and 
(Erdős et al. 1980; Bondy and Murty 2008, p. 379). The name refers
to the resemblance of this construction to the Greek letter theta.
For a simple graph, the three paths are distinct, their lengths are positive integers,
and at most one of 
,
, and 
can equal 1. The resulting graph
has 
vertices
and 
edges.
Its two common endpoints have vertex degree 3, while
every other graph vertex has vertex
degree 2.
Each pair of paths forms a graph cycle, giving three cycles of lengths 
, 
,
and 
. Therefore the theta graph is a bipartite graph iff 
, 
, and 
all have the same parity. The case
is the complete
bipartite graph 
.
