The Path Avoiding Forbidden Pairs problem (PAFP) asks whether, in a directed graph $G$ with terminals $s,t$ and a set $\mathcal{F}$ of forbidden vertex pairs, there is an $s$-$t$ path that contains at most one endpoint from each forbidden pair. We initiate the study of PAFP through a layer-based width measure. Our first focus is the union digraph $G\cup\mathcal{F}$, obtained by adding to $G$ one arc per forbidden pair, oriented according to a fixed reachability-compatible order. Let the BFS layer $L_d$ be all vertices at directed shortest-path distance $d$ from $s$, where the BFS-width from $s$ is $\max_d |L_d|$. We show if $G\cup\mathcal{F}$ has BFS-width $b$ from $s$ and only $β$ arcs going from a later BFS layer to an earlier one, then PAFP is FPT parameterized by $b+β$. The backward-arc hypothesis is essential: we show PAFP remains NP-complete when the union digraph is a DAG with BFS-width 2. We also show if the input DAG has BFS-width at most $2$ and only $k$ backward input arcs, then PAFP can be decided in $2^k |I|^{O(1)}$ time, with unrestricted forbidden pairs. This width-$2$ result is tight: inspection of a classical reduction shows NP-completeness on input DAGs of BFS-width $3$ with no backward input arcs. Moreover, we study exact-length layers in the input graph, where the $d$-th layer consists of the vertices reachable from $s$ by a directed path of length exactly $d$. For DAGs of exact-length width at most $2$, we show PAFP is polynomial-time decidable by a 2-SAT encoding of fixed-length paths. This bound is tight: the same classical reduction yields NP-completeness on DAGs of exact-length width $3$. Unlike previously known polynomial-time regimes for PAFP, which restrict the forbidden-pair set in order to obtain tractability, our two input-graph tractability results allow unrestricted forbidden pairs and input graphs with exponentially many $s$-$t$ paths.

翻译：路径避免禁止对问题（PAFP）询问：在具有终端节点$s,t$和禁止顶点对集合$\mathcal{F}$的有向图$G$中，是否存在一条$s$-$t$路径，该路径至多包含每个禁止对中的一个端点。我们通过基于层的宽度度量首次系统研究PAFP。首先关注并图$G\cup\mathcal{F}$，该图通过在$G$中为每个禁止对增加一条弧得到，弧的方向根据固定的可达性相容序确定。设BFS层$L_d$为所有与$s$有向最短路径距离为$d$的顶点，其中从$s$出发的BFS宽度定义为$\max_d |L_d|$。我们证明：若$G\cup\mathcal{F}$从$s$出发的BFS宽度为$b$，且仅有$β$条弧从后续BFS层指向先前层，则PAFP在参数$b+β$下是FPT的。后向弧假设至关重要：当并图为BFS宽度为2的有向无环图时，我们证明PAFP仍是NP完全的。进一步，若输入有向无环图的BFS宽度至多为$2$且仅有$k$条后向输入弧，则PAFP可在$2^k |I|^{O(1)}$时间内判定，且允许任意禁止对。此宽度-2结果具有紧致性：经典归约表明，在无后向输入弧但BFS宽度为$3$的输入有向无环图上，PAFP是NP完全的。此外，我们研究了输入图中的精确长度层，其中第$d$层由从$s$出发经过长度为$d$的有向路径可达的顶点构成。对于精确长度宽度至多为$2$的有向无环图，我们通过固定长度路径的2-SAT编码证明PAFP可在多项式时间内判定。该上界具有紧致性：同一经典归约表明在精确长度宽度为$3$的有向无环图上PAFP是NP完全的。与先前已知的通过限制禁止对集合实现可解性的PAFP多项式时间情形不同，我们的两个输入图可解性结果允许任意禁止对和具有指数级$s$-$t$路径的输入图。