Perfect graphs for domination games

TL;DR

This paper characterizes perfect graphs related to domination game parameters, providing polynomial recognition algorithms and identifying minimal imperfect graphs, especially within triangle-free graphs and cographs.

Contribution

It introduces recursive characterizations for b3_g- and b3_tg-perfect graphs, and identifies minimal imperfect graphs, advancing understanding of domination game perfection.

Findings

Polynomial recognition algorithm for b3_g-perfect graphs.

All minimally b3_g-imperfect graphs have domination number 2.

b3_tg-perfect graphs are exactly b1;b4;2P_3-free cographs.

Abstract

Let $\gamma_g(G)$ and $\gamma_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $\gamma_g$-perfect (resp. $\gamma_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $\gamma_g(F)=\gamma(F)$ (resp. $\gamma_{tg}(F)=\gamma_t(F)$). A recursive characterization of $\gamma_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $\gamma_g$-perfect graphs. It is proved that every minimally $\gamma_g$-imperfect graph has domination number $2$. All minimally $\gamma_g$-imperfect triangle-free graphs are determined. It is also proved that $\gamma_{tg}$-perfect graphs are precisely $\overline{2P_3}$-free cographs.