Communications in Combinatorics and Optimization (Jan 2018)
Lower bounds on the signed (total) $k$-domination number depending on the clique number
Abstract
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.
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