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 $\unicode[STIX]{x1D6FE}_{g}(G)=n/2$.

Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.

In this paper, we characterise the quasi-strong endomorphisms of the join of split graphs. We give conditions under which the quasi-strong endomorphisms of the join of split graphs form a monoid.