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 numbers. We characterize bipartite graphs with both total and Grundy total dominations 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 bipartite graphs with \(\gamma _{t}(G)=\gamma _{gr}^t(G)=6\) proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.

如果图G中没有孤立顶点的顶点序列（（v_1，\ ldots，v_k）\）称为总支配序列，如果序列中的每个顶点\（v_i \）完全支配了至少一个不完全顶点由\（\ {v_1，\ ldots，v_ {i-1} \} \）和\（\ {v_1，\ ldots，v_k \} \）主导的G的总主导集合。最短的此类序列的长度是G（\（\ gamma _ {t}（G）\））的总支配数，而最长的此类序列的长度是G（\（\伽马_ {gr} ^ t（G）\））。在本文中，我们研究具有相同总数和Grundy总控制数的图。我们对总和格朗迪总支配数均等于4的二部图进行刻画，并证明不存在具有\（\ gamma _ {t}（G）= \ gamma _ {gr} ^ t（G）的相连弦图G = 4 \）。该论文的主要结果是通过建立一个令人惊讶的对应关系证明了\（\ gamma _ {t}（G）= \ gamma _ {gr} ^ t（G）= 6 \）的二部图的刻画。这样的图和某些有限射影平面存在的经典但仍未解决的问题。