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 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

Some Bounds on the Double Domination of Signed Generalized Petersen Graphs and Signed I-Graphs

In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $\Sigma = (G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset $D \subseteq V(\Sigma)$ is a double dominating set of $\Sigma$ if it satisfies the following conditions: (i) $D$ is a double dominating set of $G$, and (ii) $\Sigma[D:V \setminus D]$ is balanced, where $\Sigma[D:V \setminus D]$ is the subgraph of $\Sigma$ induced by the edges of $\Sigma$ with one end point in $D$ and the other end point in $V \setminus D$. The cardinality of a minimum double dominating set of $\Sigma$ is the double domination number $\gamma_{\times 2}(\Sigma)$. In this paper, we give bounds for the double domination number of signed cubic graphs. We also obtain some bounds on the double domination number of signed generalized Petersen graphs and signed I-graphs.Comment: 13 page

Remarks on minus (signed) total domination in graphs

Author name used in this publication: T.C.E. ChengAuthor name also used in this publication: E.F. Shan2007-2008 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe