Resolve the remaining parameterised-complexity cases for (α,β)-Temporal Dominating Set

Determine the parameterised complexity of (0,β)-Temporal Dominating Set with respect to temporal modular-width and temporal cliquewidth when β is any positive constant; of (1,β)-Temporal Dominating Set with respect to temporal modular-width when β is any non-positive constant; and of (1,β)-Temporal Dominating Set with respect to temporal cliquewidth when β is a negative constant.

Background

The paper introduces (α,β)-Temporal Dominating Set, a temporal generalisation of (α,β)-Dominating Set that includes Temporal Dominating Set as the case α=0 and β=1. Its parameterised complexity is analysed with respect to temporal neighbourhood diversity (TND), temporal modular-width (TMW), and temporal cliquewidth (TCW).

The results establish fixed-parameter tractability for all α and β with respect to TND, W[1]-hardness for several TMW and TCW cases, and para-NP-hardness for TCW when (α,β)=(0,1) or (1,0). The cited conclusion identifies the parameter and parameter-value combinations that remain unresolved: positive constant β with α=0 under TMW or TCW; non-positive constant β with α=1 under TMW; and negative constant β with α=1 under TCW.

References

A natural next step is to solve the remaining open cases from Table \ref{tab:resultsTable}, specifically: $(0,\beta)$-tds by TMW and TCW when $\beta$ is any positive constant; $(1,\beta)$-tds by TMW when $\beta$ is any non-positive constant; and $(1,\beta)$-tds by TCW, when $\beta$ is a negative constant.

— The parameterised complexity of generalised temporal domination on temporal graphs with modular structure  (2609.26366 - Enright et al., 22 Sep 2026) in Section 5, “Conclusion and open problems,” following Table 1 (the summary table of parameterised complexity results)