@article{Yang2023, 
author = {Jian Yang and Yuefen Chen and Zhiqiang Li},
title = {Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {8},
pages = {17702-17718},
keywords = {tree, star, domination number, weak Roman domination number},
url = {https://www.sciopen.com/article/10.3934/math.2023904},
doi = {10.3934/math.2023904},
abstract = {Let    G  =  (  V  ,  E  ) be a simple graph with vertex set    V and edge set    E, and let    f be a function    f  :  V  ↦  {  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  )  >  0 such that the function        f          u        :  V  ↦  {  0  ,  1  ,  2  }, defined by        f          u        (  u  )  =  1,        f          u        (  v  )  =  f  (  v  )  −  1 and        f          u        (  w  )  =  f  (  w  ) if    w  ∈  V  −  {  u  ,  v  }, has no undefended vertex. The weight of    f is    w  (  f  )  =      ∑          v      ∈      V        f  (  v  ). The weak Roman domination number, denoted        γ          r        (  G  ), is the minimum weight of a WRDF in G. The domination number, denoted    γ  (  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 (       γ          r        (  T  )  =  γ  (  T  )  +  1) by recursion and construction.}
}