The variety of domination games

TL;DR

This paper introduces three new domination games on graphs, analyzes their properties, establishes a hierarchy among five such games, and explores their relationships with existing graph invariants.

Contribution

It defines Z-, L-, and LL-domination games, proves key principles for them, and compares their outcomes with existing domination parameters, expanding the theory of graph domination games.

Findings

Outcome differences are at most one depending on the starting player.

Hierarchy of the five domination games is established.

Values for paths are determined up to a small constant.

Abstract

Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than $6/7$ of the order of a graph.