A labelling of a graph is an assignment of labels to its vertex or edge sets (or both), subject to certain conditions, a well established concept. A labelling of a graph G of order n is termed a numbering when the set of integers {1,...,n} is used to label the vertices of G distinctly. A 2-path (a path with three vertices) in a vertex-numbered graph is said to be valid if the number of its middle vertex is smaller than the numbers of its endpoints. The problem of finding a vertex numbering of a given graph that optimises the number of induced valid 2-paths is studied, which is conjectured to be in the NP-hard class. The reported results for several graph classes show that apparently there are not one or more numbering patterns applicable to different classes of graphs, which requires the development of a specific numbering for each graph class under study.

翻译：图的标号是其顶点或边集（或两者）在特定条件下的一种赋值，这是一个成熟的概念。对于阶为n的图G，当使用整数集{1,…,n}对G的顶点进行互异编号时，该标号称为一个编号。在顶点编号图中，一条2-路径（具有三个顶点的路径）被称为有效的，如果其中间顶点的编号小于其两个端点的编号。本文研究了在给定图中寻找优化诱导有效2-路径数量的顶点编号问题，并猜想该问题属于NP难类。针对几类图的研究结果表明，显然不存在一种或多种可适用于不同图类的编号模式，这要求为所研究的每个图类开发特定的编号方法。