We introduce the graph theoretical parameter of edge treewidth. This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of edge-treewidth. We first observe that edge-treewidth does not enjoy any closeness properties under the known partial ordering relations on graphs. We introduce a variant of the topological minor relation, namely, the weak topological minor relation and we prove that edge-treewidth is closed under weak topological minors. Based on this new relation we are able to provide universal obstructions for edge-treewidth. The proofs are based on the fact that edge-treewidth of a graph is parametetrically equivalent with the maximum over the treewidth and the maximum degree of the blocks of the graph. We also prove that deciding whether the edge-treewidth of a graph is at most k is an NP-complete problem.

翻译：我们引入了边缘树枝的图形理论参数。 这个参数以树形相似的切树枝类比或树枝边象为自然方式发生。 我们研究了边- 树枝的组合特性。 我们首先观察到, 边- 树枝在图形上已知的部分顺序关系中不具有任何近距离属性。 我们引入了一个地形小关系变量, 即: 微弱的地形小关系, 我们证明, 边- 树枝在弱小的地形小群落下是封闭的。 基于这个新关系, 我们能够为边- 树枝提供普遍障碍。 证据的依据是, 图表边- 树枝与树枝的最大宽度和图形区块的最大度等同。 我们还证明, 确定一个图表的边缘- 树枝是否在 k 最多 k 之间是一个 NP 完整的问题 。