Article in press
Authors:
Jishnu Sen
National Institute of Technology Karnataka, Surathkal
Mangalore, Karnataka 575025
email: senjishnu5@gmail.com
Srinivasa Rao Kola
National Institute of Technology Karnataka, Surathkal
Mangalore, Karnataka 575025
email: srinu.iitkgp@gmail.com
Title:
Critical aspects in broadcast domination
PDFSource:
Discussiones Mathematicae Graph Theory
Received: 2022-12-10 , Revised: 2023-06-22 , Accepted: 2023-06-24 , Available online: 2023-07-19 , https://doi.org/10.7151/dmgt.2506
Abstract:
A dominating broadcast labeling of a graph $G$ is a function $f : V(G)
\rightarrow \lbrace 0, 1, 2, \dots ,\text{diam}(G)\rbrace$ such that $f(v)
\leqslant e(v)$ for all $v \in V(G)$ and
$$
\bigcup_{\substack{v \in V(G)\\ {\small f(v) > 0}}} [\lbrace u \in V(G) : d(u,v)
\leqslant f(v) \rbrace]=V(G),
$$
where $e(v)$ is the eccentricity of $v$. The cost of $f$ is $\sum_{v \in V(G)}
f(v)$. The minimum of costs over all the dominating broadcast labelings of $G$
is called the broadcast domination number $\gamma_{b}(G)$ of $G$. In this paper,
we introduce the critical aspects in broadcast domination and study it with
respect to edge deletion and edge addition. A graph $G$ is said to be
$k$-$\gamma_{b}^{+}$-edge-critical ($k$-$\gamma_{b}^{-}$-edge-critical) if
$\gamma_{b}(G-e) > \gamma_{b}(G)$, for every edge $e \in E(G)$ (if
$\gamma_{b}(G+e) < \gamma_{b}(G)$, for every edge $e \notin E(G)$), where
$\gamma_{b}(G)=k$. We give a necessary and sufficient condition for a graph to
be $k$-$\gamma_{b}^{+}$-edge-critical. We characterize
$k$-$\gamma_{b}^{-}$-edge-critical graphs for $k=1, 2$, and give necessary
conditions of the same for $k \geqslant 3$. Further, we define the broadcast
bondage number and the broadcast reinforcement number of a graph, and give tight
upper bounds for them.
Keywords:
dominating broadcast labeling, broadcast domination number, critical graph, bondage number, reinforcement number
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