标题： 图的绕道覆盖数和束带覆盖数 显示英文标题

标题： The detour covering number and cummerbund covering number of a graph

摘要： 我们引入了关于图的几个新概念，并研究了它们的基本性质。 图中最长路径称为绕道，最长圈称为腰带。 图的绕道覆盖数是绕道所包含的顶点个数。 如果图中的每个顶点都位于一条绕道上，则称该图为绕道覆盖图。 腰带覆盖数和腰带覆盖图的定义类似。 一些主要结果如下： (1) 确保一个图成为腰带覆盖图或绕道覆盖图的最小度数条件和禁止子图条件。 (2) 具有连通性或围长条件的图的最小腰带覆盖数和最小绕道覆盖数。 (3)$2$- 连通二分图的最小腰带覆盖数和极值图。 显示英文摘要

摘要： We introduce several new concepts about graphs and investigate their basic properties. A longest path in a graph is called a detour and a longest cycle is called a cummerbund. The detour covering number of a graph is the number of vertices that lie in a detour. A graph is said to be detour covered if every vertex lies in a detour. The cummerbund covering number and cummerbund covered graphs are defined similarly. Some of the main results are as follows. (1) Minimum degree and forbidden subgraph conditions that ensure a graph to be cummerbund covered or detour covered. (2) The minimum cummerbund covering number and minimum detour covering number of a graph with connectivity or girth conditions. (3) The minimum cummerbund covering number of a $2$-connected bipartite graph and the extremal graphs.