We obtain structure theorems for graphs excluding a fan (a path with a universal vertex) or a dipole ($K_{2,k}$) as a topological minor. The corresponding decompositions can be computed in FPT linear time. This is motivated by the study of a graph parameter we call treebandwidth which extends the graph parameter bandwidth by replacing the linear layout by a rooted tree such that neighbours in the graph are in ancestor-descendant relation in the tree. We deduce an approximation algorithm for treebandwidth running in FPT linear time from our structure theorems. We complement this result with a precise characterisation of the parameterised complexity of computing the parameter exactly.

翻译：我们针对排除扇（具有一个通用顶点的路径）或偶极子（$K_{2,k}$）作为拓扑子式的图，获得了结构定理。相应的分解可以在FPT线性时间内计算。这一研究动机源于我们对称为树带宽的图参数的研究，该参数通过将线性布局替换为根树（使得图中的邻接点在树中呈祖先-后代关系）来扩展带宽这一图参数。基于我们的结构定理，我们推导出一种在FPT线性时间内运行的树带宽近似算法。我们通过精确刻画计算该参数的确切参数化复杂度，对此结果进行了补充。