We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges (s can be seen as the speed of the robber). Both parts have complete knowledge of the opponent's moves and the cops win when they occupy all edges incident to the robbers position. We introduce the capture cost on G against a robber of speed s. This defines a hierarchy of invariants, namely ${\delta }_{\mathrm{e}}^1,{\delta }_{\mathrm{e}}^2,\dots ,{\delta }_{\mathrm{e}}^{\infty }$, where ${\delta }_{\mathrm{e}}^{\infty }$ is an edge-analogue of the admissibility graph invariant, namely the edge-admissibility of a graph. We prove that the problem asking whether ${\delta }_{\mathrm{e}}^s(G)\le k$, is polynomially solvable when $s\in \{1,2,3,\infty \}$ while, otherwise, it is NP-complete. Our main result is a structural theorem for graphs of bounded edge-admissibility. We prove that every graph of edge-admissibility at most k can be constructed using $(\le k)$-edge-sums, starting from graphs whose all vertices, except possibly from one, have degree at most k. Our structural result is approximately tight in the sense that graphs generated by this construction always have edge-admissibility at most $2k-1$. Our proofs are based on a precise structural characterization of the graphs that do not contain ${\theta }_r$ as an immersion, where ${\theta }_r$ is the graph on two vertices and r parallel edges.

我们考虑一个警察和强盗游戏，其中警察阻塞了图的边缘，而强盗占据了其顶点。在游戏的每个回合中，警察都会在强盗被迫移动到另一个遍历最多s个畅通无阻的边缘的顶点后立即选择要遮挡的一组边缘（可以将s视为强盗的速度）。这两个部分都完全了解对手的动作，当警察占据与强盗位置有关的所有优势时，他们就会获胜。我们介绍了G相对于强盗s的捕获成本。这定义了不变式的层次结构，即${\delta }_{\mathrm{Ë}}^{\mathrm{1个}}，{\delta }_{\mathrm{Ë}}^2，\dots ，{\delta }_{\mathrm{Ë}}^{\infty }$，在哪里 ${\delta }_{\mathrm{Ë}}^{\infty }$是可容许性图不变性的边缘模拟，即图的可容许性。我们证明问题是${\delta }_{\mathrm{Ë}}^s（G）\le ķ$，在以下情况下可以多项式求解 $s\in \{\mathrm{1个}，2，3，\infty \}$否则，它是NP -complete。我们的主要结果是有界边可容许性图的结构定理。我们证明可以使用以下公式构造最多k个边缘可容许性的每个图$（\le ķ）$-edge-sums-从所有顶点（除可能一个顶点之外）的所有顶点都具有度k的图开始。从这种构造生成的图形最多始终具有边缘可容许性的意义上说，我们的结构结果大约是紧密的$2ķ-\mathrm{1个}$。我们的证明基于不包含图的图的精确结构特征${\theta }_{\mathrm{[R}}$ 作为沉浸， ${\theta }_{\mathrm{[R}}$是在两个顶点和r个平行边上的图。