On the signed strong total Roman domination number of graphs
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Abstract
Let G=(V,E) be a finite and simple graph of order n and maximum
degree Δ. A signed strong total Roman dominating function on
a graph G is a function f:V(G)→{−1,1,2,…,⌈Δ2⌉+1} satisfying the condition that (i) for
every vertex v of G, f(N(v))=∑u∈N(v)f(u)≥1, where
N(v) is the open neighborhood of v and (ii) every vertex v for
which f(v)=−1 is adjacent to at least one vertex
w for which f(w)≥1+⌈12|N(w)∩V−1|⌉, where
V−1={v∈V:f(v)=−1}.
The minimum of the
values ω(f)=∑v∈Vf(v), taken over all signed strong
total Roman dominating functions f of G, is called the signed strong total
Roman domination number of G and is denoted by γssTR(G).
In this paper, we initiate signed strong total Roman domination number of a graph and give
several bounds for this parameter. Then, among other results, we determine the signed strong total Roman domination
number of special classes of graphs.
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