@article{Al Subaiei2023, 
author = {Bana Al Subaiei and Ahlam AlMulhim and Abolape Deborah Akwu},
title = {Vertex-edge perfect Roman domination number},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {9},
pages = {21472-21483},
keywords = {vertex-edge perfect domination number, trees, cycles, Petersen graph, bipartite graph},
url = {https://www.sciopen.com/article/10.3934/math.20231094},
doi = {10.3934/math.20231094},
abstract = {A vertex-edge perfect Roman dominating function on a graph    G  =  (  V  ,  E  ) (denoted by ve-PRDF) is a function    f  :  V      (    G    )    ⟶  {  0  ,  1  ,  2  } such that for every edge    u  v  ∈  E,    max  {  f  (  u  )  ,  f  (  v  )  }  ≠  0, or    u is adjacent to exactly one neighbor    w such that    f  (  w  )  =  2, or    v is adjacent to exactly one neighbor    w such that    f  (  w  )  =  2. The weight of a ve-PRDF on    G is the sum    w  (  f  )  =      ∑          v      ∈      V        f  (  v  ). The vertex-edge perfect Roman domination number of    G (denoted by        γ          v      e      R              p        (  G  )) is the minimum weight of a ve-PRDF on    G. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree    T, we give upper and lower bounds for        γ          v      e      R              p        (  T  ) in terms of the order    n,    l leaves and    s support vertices. Lastly, we determine        γ          v      e      R              p        (  G  ) for Petersen, cycle and Flower snark graphs.}
}