Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1

Abstract

<abstract><p>Let $ G = (V, E) $ be a simple graph with vertex set $ V $ and edge set $ E $, and let $ f $ be a function $ f:V\mapsto \{0, 1, 2\} $. A vertex $ u $ with $ f(u) = 0 $ is said to be undefended with respect to $ f $ if it is not adjacent to a vertex with positive weight. The function $ f $ is a weak Roman dominating function (WRDF) if each vertex $ u $ with $ f(u) = 0 $ is adjacent to a vertex $ v $ with $ f(v) &gt; 0 $ such that the function $ f_{u}:V\mapsto \{0, 1, 2\} $, defined by $ f_{u}(u) = 1 $, $ f_{u}(v) = f(v)-1 $ and $ f_{u}(w) = f(w) $ if $ w\in V-\{u, v\} $, has no undefended vertex. The weight of $ f $ is $ w(f) = \sum_{v\in V}f(v) $. The weak Roman domination number, denoted $ \gamma_{r}(G) $, is the minimum weight of a WRDF in G. The domination number, denoted $ \gamma(G) $, is the minimum cardinality of a dominating set in $ G $. In this paper, we give some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 ($ \gamma_{r}(T) = \gamma(T)+1 $) by recursion and construction.</p></abstract>