On graphs with equal total domination and Grundy total domination number

TL;DR

This paper investigates graphs where the total domination number equals the Grundy total domination number, providing characterizations for bipartite graphs with these properties and linking the problem to the existence of finite projective planes.

Contribution

It characterizes bipartite graphs with equal total and Grundy total domination numbers for specific values and connects the problem to an open question in finite geometry.

Findings

Bipartite graphs with total and Grundy total domination number 4 are characterized.

No connected chordal graph has total and Grundy total domination number 4.

Regular bipartite graphs with these parameters relate to finite projective planes.

Abstract

A sequence $(v_1,\ldots ,v_k)$ of vertices in a graph $G$ without isolated vertices is called a total dominating sequence if every vertex $v_i$ in the sequence totally dominates at least one vertex that was not totally dominated by $\{v_1,\ldots , v_{i-1}\}$ and $\{v_1,\ldots ,v_k\}$ is a total dominating set of $G$. The length of a shortest such sequence is the total domination number of G ($\gamma_t(G)$), while the length of a longest such sequence is the Grundy total domination number of $G$ ($\gamma_{gr}^t(G)$). In this paper we study graphs with equal total and Grundy total domination number. We characterize bipartite graphs with both total and Grundy total domination number equal to 4, and show that there is no connected chordal graph $G$ with $\gamma_t(G)=\gamma_{gr}^t(G)=4$. The main result of the paper is a characterization of regular bipartite graphs with $\gamma_t(G)=\gamma_{gr}^t(G)=6$ proved by establishing a surprising correspondence between existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.