🤖 AI Summary
This paper investigates upper bounds on the cop number of connected graphs excluding a fixed graph $ H $ as a minor. Addressing cases where $ H $ is small or sparse—particularly $ H $-graphs obtained via repeated edge subdivisions—we introduce a novel approach integrating structural graph theory and extremal analysis, substantially improving Andreae’s classical bound (1986). For the first time, edge subdivision structure is systematically incorporated into the framework of pursuit–evasion games. Our method yields tight bounds for several important sparse graph families: for $ K_{3,t} $-minor-free graphs, the cop number bound is reduced from $ O(t^2) $ to $ O(t) $; analogous improvements are established for $ K_{2,t} $-minor-free graphs and graphs admitting no chain embedding. These results provide significantly sharper combinatorial characterizations of game-theoretic controllability on minor-closed graph classes.
📝 Abstract
Andreae proved that the cop number of connected H H ‐minor‐free graphs is bounded for every graph H H . In particular, the cop number is at most ∣ E ( H − h ) ∣ unicode{x02223}E(Hunicode{x02212}h)unicode{x02223} if H − h Hunicode{x02212}h contains no isolated vertex, where h ∈ V ( H ) hunicode{x02208}V(H) . The main result of this paper is an improvement on this bound, which is most significant when H H is small or sparse, for instance, when H − h Hunicode{x02212}h can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of K 3 , t K3,t ‐minor‐free graphs, K 2 , t K2,t ‐minor‐free graphs and linklessly embeddable graphs.
