### A New Upper Bound on the Total Domination Number of a Graph

Michael A. Henning, Anders Yeo
2007 Electronic Journal of Combinatorics
A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than $2$ and we let ${\cal L}$ be the set of all large vertices of $G$. Let $P$ be any component of $G - {\cal more » ... onent of$G - {\cal L}$; it is a path. If$|P| \equiv 0 \, ( {\rm mod} \, 4)$and either the two ends of$P$are adjacent in$G$to the same large vertex or the two ends of$P$are adjacent to different, but adjacent, large vertices in$G$, we call$P$a$0$-path. If$|P| \ge 5$and$|P| \equiv 1 \, ( {\rm mod} \, 4)$with the two ends of$P$adjacent in$G$to the same large vertex, we call$P$a$1$-path. If$|P| \equiv 3 \, ( {\rm mod} \, 4)$, we call$P$a$3$-path. For$i \in \{0,1,3\}$, we denote the number of$i$-paths in$G$by$p_i$. We show that the total domination number of$G$is at most$(n + p_0 + p_1 + p_3)/2$. This result generalizes a result shown in several manuscripts (see, for example, J. Graph Theory 46 (2004), 207–210) which states that if$G$is a graph of order$n$with minimum degree at least three, then the total domination of$G$is at most$n/2$. It also generalizes a result by Lam and Wei stating that if$G$is a graph of order$n$with minimum degree at least two and with no degree-$2$vertex adjacent to two other degree-$2$vertices, then the total domination of$G$is at most$n/2\$.