For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the zero forcing polynomial of every $n$-vertex graph should be coefficientwise dominated by that of $P_n$. We prove this path-extremal conjecture for distance-hereditary graphs. This extends the previously known tree case to a much larger class that includes, in particular, all trees and all cographs. We then use canonical split decomposition to push the argument one step beyond the distance-hereditary setting. Specifically, we show that if a split-prime graph $H$ and all of its induced subgraphs are path-extremal, then every connected graph whose canonical split decomposition has a unique prime bag whose label graph is isomorphic to $H$ is also path-extremal. As a corollary, for each fixed $m$, if every induced subgraph of every split-prime graph on at most $m$ vertices is path-extremal, then so is every connected graph whose canonical split decomposition has a unique prime bag of size at most $m$. Thus, on these classes, the conjecture reduces to a finite verification problem on bounded-order prime cores. Our proofs combine two counting mechanisms for non-forcing sets -- fort obstructions arising from twin pairs and a leaf recurrence -- with the accessibility description of graph-labelled trees in the canonical split decomposition. This yields a new positive instance of the path-extremal conjecture and identifies a natural structural frontier for further progress.

翻译：对于$n$个顶点的图$G$，令$z(G;k)$表示大小为$k$的零强制集的个数。Boyer等人的猜想声称，在所有$n$个顶点的图中，路径图$P_n$在系数意义下最大化这些数值；等价地，每个$n$顶点图的零强制多项式应在系数上被$P_n$的零强制多项式控制。我们证明了距离遗传图满足该路径极值猜想。这将其已知的树情形推广到包含所有树和所有余图在内的更广泛的图类。然后我们利用典范分裂分解将论证推进到距离遗传设置之外。具体而言，我们证明：若分裂素图$H$及其所有诱导子图都是路径极值的，则每个典范分裂分解具有唯一标签图同构于$H$的素袋的连通图也是路径极值的。作为推论，对每个固定$m$，若每个至多$m$个顶点的分裂素图的所有诱导子图都是路径极值的，则每个典范分裂分解中具有唯一大小至多为$m$的素袋的连通图也是路径极值的。因此，在这些图类上，该猜想可归约为有界阶素核上的有限验证问题。我们的证明将两种非强制集的计数机制——源自孪生对的堡垒障碍与叶递归——与典范分裂分解中图标记树的可达性描述相结合。这为路径极值猜想提供了一个新的正面实例，并指出了进一步研究的一个自然结构前沿。