Lower bounds on the signed (total) $k$-domination number

Document Type: Original paper

Author

RWTH Aachen University

Abstract

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating function
is a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)
for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values
$sum_{vin V(G)}f(v)$, 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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