A 二部グラフ is a special kind of graph that is divided into two distinct sets of vertices. In this type of graph, the edges connect vertices from one set to those in the other set, but there are no edges connecting vertices within the same set. This structure is often used to model relationships between two different types of entities.
形式的には、二部グラフはグラフ G = (U, V, E) と定義され、U と V は互いに交わらない2つの頂点集合であり、E はエッジの集合です。各エッジは、集合 U の頂点と集合 V の頂点をつなぎます。二部グラフの例としては、マッチメイキングのシナリオを表すグラフがあり、一方の集合は求職者を表し、もう一方の集合は求人を表します。
二部グラフは、さまざまな分野で重要な役割を果たしています。 コンピュータ科学, social ネットワーク分析, and 運用研究. They are particularly useful in modeling situations where two different classes of objects interact. For example, in レコメンデーションシステム, one set could represent users and the other set could represent items to recommend.
In terms of properties, a bipartite graph can be characterized by the absence of odd-length cycles. This means that if a bipartite graph is colored using two colors, no two adjacent vertices will share the same color. This property is useful in graph algorithms and helps in determining if a graph is bipartite using techniques like breadth-first search (BFS) or 深さ優先探索 (DFS)。