区间图Total-罗马控制性质研究
Total-Roman Domination Property on Interval Graphs

Total-罗马控制函数是函数f:V(G)→{0,1,2}，满足条件：1) 对G中任意函数值f(u)=0的顶点u，至少存在一个邻居v使得函数值f(v)=2；2) 由控制集{k|f(k)≥1且k∈V(G)}诱导的子图没有孤立点存在。结合3阶及以上区间图，本文主要探索了基于total-罗马对的定理，研究了团和路径的不同结合图类中total-罗马控制数等内容，以示例辅助理解，证明了任意阶团的total-罗马控制数为3、相交团的并的total-罗马控制数不超过4等性质。
Total-Roman Domination Function is the function f:V(G)→{0,1,2}, which satisfies the following two conditions: 1) for any vertex u with f(u)=0, there is at least one neighbor v with f(v)=2；2) No outliers exist for a subgraph induced by a control set {k|f(k)≥1 and k∈V(G)}. Combined with interval graphs of order 3 and above, this paper mainly explores the Total-Roman control number based on the theorem of Total-Roman pairs, studies the Total-Roman control number of different associative graph classes of groups and paths, and proves that the total-Roman control number of any order group is 3, and the Total-Roman control number of union of intersecting groups is not more than 4 and other properties.