[Turkmath:6319] Hacettepe Üniversitesi Genel Semineri-Selim Bahadır

[Asli Pekcan]

Sayın Liste Üyeleri,
Hacettepe Üniversitesi Matematik Bölümü genel seminerleri kapsamında, 20 Aralık
2023 tarihinde saat 15:30'da, bölümümüz Yaşar Ataman toplantı salonunda,
Ankara Yıldırım Beyazıt Üniversitesi'nden Selim Bahadır'ın vereceği ''On
graphs all of whose total dominating sequences have the same length''
başlıklı konuşmaya
ilgilenen herkesi bekleriz.

Saygılarımla,

Prof. Dr. Aslı Pekcan Yıldız
Seminer Koordinatörü
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*Konuşma Başlığı: * On graphs all of whose total dominating sequences have
the same length
*Konuşma Özeti*: A sequence of vertices in a graph $G$ without isolated
vertices is called a total dominating sequence if every vertex $v$ in the
sequence has a neighbor which is adjacent to no vertex preceding $v$ in the
sequence, and at the end every vertex of $G$ has at least one neighbor in
the sequence. Minimum and maximum lengths of a total dominating sequence is
the total domination number of $G$ (denoted by $\gamma_t(G)$) and the
Grundy total domination number of $G$ (denoted by $\gamma_{gr}^t(G)$),
respectively.

In this paper, we study graphs where all total dominating sequences
have the same length. For every positive integer $k$, we call $G$ a
total $k$-uniform graph if every total dominating sequence of $G$ is
of length $k$, that is, $\gamma_t(G)=\gamma_{gr}^t(G)=k$. We prove
that there is no total $k$-uniform graph when $k$ is odd. In addition,
we present a total 4-uniform graph which stands as a counterexample
for a conjecture by Gologranc et al. 2021 and provide a connected
total 8-uniform graph. Moreover, we prove that every total
$k$-uniform, connected and false twin-free graph is regular for every
even $k$. We also show that there is no total $k$-uniform chordal
connected graph with $k\geq 4$ and characterize all total $k$-uniform
chordal graphs.
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