vertexmenge

Vertexmenge is the German term for the vertex set of a graph. In graph theory, a graph G is defined by a set V of vertices (or nodes) and a set E of edges (or arcs) that connect pairs of vertices. The vertexmenge V(G) consists of all vertices that are part of the graph G. It is a fundamental concept because many properties of a graph are defined in terms of its vertexmenge, such as the graph's order, which is the cardinality |V(G)|, indicating how many vertices the graph contains.

The vertexmenge can be finite or infinite. In a finite simple graph, each vertex is unique, and the vertexmenge is typically denoted by a capital letter, e.g., V or V(G). For directed graphs, each vertex is still part of the vertexmenge, although each edge has a direction, which affects traversal but not the set of vertices themselves.

Operations on vertexmenge include subgraph extraction, where a subgraph H of G is defined by a subset of V(G) and a subset of edges E(H) where each edge connects vertices in the subset. The complement of a graph also involves the vertexmenge, as the complement graph retains the same vertexmenge but with a different edge set.

In network analysis, the vertexmenge often corresponds to distinct entities such as actors in a social network or routers in a computer network. Understanding the structure and properties of the vertexmenge is essential for tasks such as connectivity analysis, shortest path computations, and graph coloring.