Abstract

We consider the class T of 2-connected non-planar K 3 , 3 -subdivision-free graphs that are embeddable in the torus. We show that any graph in T admits a unique decomposition as a basic toroidal graph (the toroidal core) where the edges are replaced by two-pole networks constructed from 2-connected planar graphs. The structure theorem provides a practical algorithm to recognize toroidal graphs with no K 3 , 3 -subdivisions in linear time. Labelled toroidal cores are enumerated, using matching polynomials of cycle graphs. As a result, we enumerate labelled graphs in T having vertex degree at least two or three, according to their number of vertices and edges. We also show that the number m of edges of graphs in T satisfies the bound m ⩽ 3 n - 6 , for n ⩾ 6 vertices, n ≠ 8 .

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