We investigate the domination game and the game domination number $\unicode[STIX]{x1D6FE}_{g}$ in the class of split graphs. We prove that $\unicode[STIX]{x1D6FE}_{g}(G)\leq n/2$ for any isolate-free $n$-vertex split graph $G$, thus strengthening the conjectured $3n/5$ general bound and supporting Rall’s $\lceil n/2\rceil$-conjecture. We also characterise split graphs of even order with $\unicode[STIX]{x1D6FE}_{g}(G)=n/2$.

We study the (1:b) Maker–Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to prevent him from doing so. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players.

To this end, we prove an extension of a theorem of Gallai, Hasse, Roy and Vitaver about graph orientations without long directed simple paths.

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of r revolutionaries tries to hold an unguarded meeting consisting of m revolutionaries. A team of s spies wants to prevent this forever. For given r and m, the minimum number of spies required to win on a graph G is the spy number σ(G,r,m). We present asymptotic results for the game played on random graphs G(n,p) for a large range of p = p(n), r=r(n), and m=m(n). The behaviour of the spy number is analysed completely for dense graphs (that is, graphs with average degree at least n1/2+ε for some ε > 0). For sparser graphs, some bounds are provided.