Grundy double domination number: bounds, graph operations, and efficient computation for $P_4$-tidy graphs

TL;DR

This paper investigates the Grundy double domination number in graphs, establishing bounds, analyzing graph operations, and providing a linear-time algorithm for $P_4$-tidy graphs, thus addressing an open computational problem.

Contribution

It introduces tight bounds for the GDDN, explores GDDSs under graph modifications, and presents a linear-time algorithm for $P_4$-tidy graphs, solving a previously open problem.

Findings

Established tight bounds for GDDN.

Described GDDSs for vertex-removed graphs and graph joins.

Proved linear-time computation of GDDN for $P_4$-tidy graphs.

Abstract

Inspired by graph domination games, various domination-type vertex sequences have been introduced, including the Grundy double dominating sequence (GDDS) of a graph and its associated parameter, the Grundy double domination number (GDDN). The decision version of the problem of computing the GDDN is known to be NP-complete, even when restricted to split graphs and bipartite graphs. In this paper, we establish general tight bounds for the GDDN. We also describe GDDSs for vertex-removed graphs and for the join of two graphs. Applying these results, we prove that computing the GDDN is linear for $P_4$-tidy graphs, thereby solving an open problem previously posed for cographs by B. Bre\v{s}ar et al. in [Bre\v{s}ar, B., Pandey, A., and Sharma, G. (2022). Computational aspects of some vertex sequences of grundy domination-type. Indian J. Discrete Math., 8:21-38].