A proper circular‑arc (PCA) model is a pair $M=(C,A)$ where $C$ is a circle and $A$ is a family of inclusion‑free arcs whose endpoints are pairwise distinct. The model represents a digraph $D$: each arc $a\in A$ yields a vertex $v(a)$, and for any pair of arcs $a,b\in A$, if the beginning point of $b$ lies on $a$, then $D$ contains the edge $v(a)\to v(b)$. For $k\ge 0$, the $k$‑th power $D^k$ keeps the same vertex set, and $v(a)\to v(b)$ belongs to $D^k$ whenever $a\neq b$ and the distance from $v(a)$ to $v(b)$ in $D$ does not exceed $k$.
A unit circular‑arc (UCA) model $U=(C,A)$ is a PCA model where all arcs have the same length $\ell+1$. If $\ell$, the circle length $c$, and all arc endpoints are integers, $U$ is a $(c,\ell)$‑CA model. For $i\ge 0$, the model $i\times U$ is obtained by replacing each arc $(s,s+\ell+1)$ with $(s,s+i\ell+1)$. If $U$ represents a digraph $D$, then $U$ is $k$‑multiplicative when, for every $0\le i\le k$, $i\times U$ represents $D^i$.
The paper presents a linear‑time algorithm that decides whether a given PCA model $M$ is equivalent to a $k$‑multiplicative UCA model (with $k$ provided as input). The algorithm either produces an equivalent $k$‑multiplicative UCA model $U$ or a negative certificate that can be authenticated in $O(|M|)$ time. The main technical contribution is a new characterization of PCA models that admit a $k$‑multiplicative UCA representation. For $k=1$, this characterization yields a simpler algorithm for the classic representation problem than previously known solutions.
Review: The linear‑time decision procedure advances the theory of circular‑arc graph representations, especially by unifying distance‑bounded multiplicative extensions, and opens new avenues for efficient graph algorithm design.