2 research outputs found
On the Jha/Pradhan/Banerjee algorithm for the secure domination number of cographs
Jha, Pradhan, and Banerjee devised a linear algorithm to compute the secure
domination number of a cograph. Here it is shown that their Lemma~2, which is
crucial for the computational complexity of the algorithm, is incomplete. An
accordingly modified lemma is proved and it is demonstrated that the complexity
of the modified algorithm remains linear
A Note on Roman \{2\}-domination problem in graphs
For a graph , a Roman -dominating function
(R2DF) has the property that for every vertex with , either there exists a neighbor , with , or
at least two neighbors having . The weight of a R2DF
is the sum , and the minimum weight of a R2DF is the
Roman -domination number . A R2DF is independent if
the set of vertices having positive function values is an independent set. The
independent Roman -domination number is the minimum
weight of an independent Roman -dominating function on . In this
paper, we show that the decision problem associated with
is NP-complete even when restricted to split graphs. We design a linear time
algorithm for computing the value of for any tree . This
answers an open problem raised by Rahmouni and Chellali [Independent Roman
-domination in graphs, Discrete Applied Mathematics 236 (2018),
408-414]. Chellali, Haynes, Hedetniemi and McRae \cite{chellali2016roman} have
showed that Roman -domination number can be computed for the class of
trees in linear time. As a generalization, we present a linear time algorithm
for solving the Roman -domination problem in block graphs