Cutting Lemma and Union Lemma for the Domination Game

TL;DR

This paper introduces the cutting and union lemmas for the domination game, providing new bounds and techniques that lead to proving a conjecture about three-legged spiders and classifying critical trees.

Contribution

It presents two novel lemmas for the domination game, proves a conjecture on three-legged spiders, and classifies critical trees up to 20 vertices.

Findings

Proved the conjecture that three-legged spiders are game domination critical.

Derived an extended cutting lemma for the domination game.

Listed all game domination critical trees with 18-20 vertices.

Abstract

Two new techniques are introduced into the theory of the domination game. The cutting lemma bounds the game domination number of a partially dominated graph with the game domination number of suitably modified partially dominated graph. The union lemma bounds the S-game domination number of a disjoint union of paths using appropriate weighting functions. Using these tools a conjecture asserting that the so-called three legged spiders are game domination critical graphs is proved. An extended cutting lemma is also derived and all game domination critical trees on 18, 19, and 20 vertices are listed.