New bounds on the signed total domination number of graphs

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turan, we present a sharp lower bound on this parameter for graphs with no complete graph of order r+1 as a subgraph. Also, we prove that n-2(s-s') is an upper bound on the signed total domination number of any tree of order n with s support vertices and s' support vertives of degree two. Moreover, we characterize all trees attainig this bound.Comment: This paper contains 11 pages and one figur

On the inverse signed total domination number in graphs

In this paper, we study the inverse signed total domination number in graphs and present new lower and upper bounds on this parameter. For example by making use of the classic theorem of Turan (1941), we present a sharp upper bound for graphs with no induced complete subgraph of order greater than two. Also, we bound this parameter for a tree in terms of its order and the number of leaves and characterize all trees attaining this bound

Lower bounds on the signed (total) kk-domination number depending on the clique number

Let $G$ be a graph with vertex set $V(G)$‎. ‎For any integer $k\ge 1$‎, ‎a signed (total) $k$-dominating function‎ ‎is a function $f‎: ‎V(G) \rightarrow‎ \{ -1, ‎1\}$ satisfying $\sum_{x\in N[v]}f(x)\ge k$ ($\sum_{x\in N(v)}f(x)\ge k$)‎ ‎for every $v\in V(G)$‎, ‎where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)\cup\{v\}$‎. ‎The minimum of the values‎ ‎$\sum_{v\in V(G)}f(v)$‎, ‎taken over all signed (total) $k$-dominating functions $f$‎, ‎is called the signed (total)‎ ‎$k$-domination number‎. ‎The clique number of a graph $G$ is the maximum cardinality of a complete subgraph of $G$‎. ‎In this note we present some new sharp lower bounds on the signed (total) $k$-domination number‎ ‎depending on the clique number of the graph‎. ‎Our results improve some known bounds

New Bounds on the Signed Total Domination Number of Graphs

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on Kr+1-free graphs for r ≥ 2. Applying the concept of total limited packing we bound the signed total domination number of G with δ(G) ≥ 3 from above by . Also, we prove that γst(T) ≤ n − 2(s − s′) for any tree T of order n, with s support vertices and s′ support vertices of degree two. Moreover, we characterize all trees attaining this bound

New Bounds on the Signed Total Domination Number of Graphs

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on $K_{r+1}$-free graphs for $r \ge 2$. Applying the concept of total limited packing we bound the signed total domination number of $G$ with $\delta (G) \ge 3$ from above by n - 2 \floor{ \frac{ 2 \rho_0 (G) + \delta - 3 }{ 2 } } . Also, we prove that $\gamma_{st} (T) \le n − 2(s − s^′ )$ for any tree $T$ of order$$ n, with $s$ support vertices and $s^′$ support vertices of degree two. Moreover, we characterize all trees attaining this bound