Almost series-parallel graphs: structure and colorability

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 K4 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” (ASP) graphs still admit a concise description. Amongst other things, their description reveals that: 1. Essentially, the 3-connected ASP graphs are the cubic graphs obtained from the 3-connected cubic graphs by replacing each vertex with a triangle (i.e. the 3-connected cubic claw-free graphs). 2. Except for K 6 , the ASP graphs are 5-colorable in polynomial time; there are 5-chromatic ASP graphs and 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 arose from a previous paper of ours:“On the colorability of graphs with forbidden minors along paths and circuits, Discrete Math. (to appear)”.