2026/05/11 by Samuel German · 1 voice
Computer Science · #cs.DM
paper · pdf · doi:10.48550/arxiv.2605.10836
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 Pn 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 Pn. 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.