  The series-parallel (SP) graphs are those containing no topological $K_{_4}$
and are considered trivial. We relax the prohibition distinguishing the SP
graphs by forbidding only embeddings of $K_{_4}$ whose edges with both ends
3-valent (skeleton hereafter) induce a graph isomorphic to certain prescribed
subgraphs of $K_{_4}$. In particular, we describe the structure of the graphs
containing no embedding of $K_{_4}$ whose skeleton is isomorphic to $P_{_3}$ or
$P_{_4}$. Such "almost series-parallel graphs" (ASP) still admit a concise
description. Amongst other things, their description reveals that:
  1. Essentially, the 3-connected ASP graphs are those obtained from the
3-connected cubic graphs by replacing each vertex with a triangle (e.g., the
3-connected claw-free graphs).
  2. Except for $K_{_6}$, the ASP graphs are 5-colorable in polynomial time.
Distinguishing between the 5-chromatic and the 4-colorable ASP graphs is
$NP$-hard.
  3. The ASP class is significantly richer than the SP class:
4-vertex-colorability, 3-edge-colorability, and Hamiltonicity are $NP$-hard for
ASP graphs.
  Our interest in such ASP graphs arises from a previous paper of ours: "{\sl
On the colorability of graphs with forbidden minors along paths and circuits},
Discrete Math. (to appear)".
