计算机问题求解-论题3-10 图的连通度 2020年11月24日
计算机问题求解 – 论题3-10 - 图的连通度 2020年11月24日
割点也是观察和衡量图连通特性的关键 :<I 问题1： 你能否解释一下它们的“连通” 怎么不一样？ 割点、割边与回路
割点也是观察和衡量图连通特性的关键 割点、割边与回路
围绕割点的若干结论 Theorem 5.1 Let y be a vertex incident with a bridge in a connected graph G.Then v is a cut-vertex of G if and only if degv≥2. Theorem 5.3 Let v be a cut-vertex in a connected graph G and let u and w be vertices in distinct components of G-v. Then v lies on every u-w path in G. Theorem 5.5 Let G be a nontrivial connected graph and let u EV(G).Ifv is a vertex that is farthest from u in G,then v is not a cut-vertex of G
围绕割点的若干结论
问题2 衡量连通的牢度”的指 标是什么？
指标1：minimum vertex-cut By a vertex-cut in a graph G,we mean a set U of vertices of G such that G-U is disconnected.A vertex- cut of minimum cardinality in G is called a minimum vertex-cut
指标1：minimum vertex-cut By a vertex-cut in a graph G, we mean a set U of vertices of G such that G - U is disconnected. A vertexcut of minimum cardinality in G is called a minimum vertex-cut
围绕割点的重要概念之一：不可分割 A graph of order at least 3 is nonseparable if and only if every two vertices lie on a common cycle ■充分条件 口显然，删除任意一点，均不能使得任意两点不连通 ■必要条件：反证法 Why? 口找到不成回路的最短uV路径P ▣令Vk.1是P中v的直接前驱节点 u,k1有回路C ■删去k1,u和V仍然相连：Q Uk=U ■令x是C和Q的第一个交点 ■构造P',Q',可以形成uX+Q'+P'的uV回路
围绕割点的重要概念之一：不可分割 A graph of order at least 3 is nonseparable if and only if every two vertices lie on a common cycle ◼ 充分条件 ❑ 显然，删除任意一点，均不能使得任意两点不连通 ◼ 必要条件：反证法 ❑ 找到不成回路的最短uv路径P ❑ 令vk-1是P中v的直接前驱节点 ◼ u, vk-1有回路C ◼ 删去vk-1 ，u和v仍然相连:Q ◼ 令x是C和Q的第一个交点 ◼ 构造P’,Q’,可以形成ux+Q’+P’的uv回路 Why？
围绕割点的重要概念之二：Block A maximal nonseparable subgraph of a graph 6 is called a block of G 问题3：极大还 是最大？
围绕割点的重要概念之二：Block A maximal nonseparable subgraph of a graph G is called a block of G. 问题3：极大还 是最大？
关于block的几个理解 Uk-1 Uk=U 问题4：从这个定理及 P 证明中能否看出，不 可分离图在外表上具 有什么共性？
问题4：从这个定理及 证明中能否看出，不 可分离图在外表上具 有什么共性？ 关于block的几个理解
关于block的几个理解 Theorem 5.8 Let R be the relation defined on the edge set of a nontrivial connected graph G by e R f,where e,f eE(G),if e f or e and f lie on a common cycle of G.Then R is an equivalence relation. 用等价关系来描述边之间的关系，到底有何用意？ 问题5：Block的“极大”特性和“等价类”有何关系？ Each subgraph of G induced by the edges in an equivalence class is in fact a block of G
关于block的几个理解 Theorem 5.8 Let R be the relation defined on the edge set of a nontrivial connected graph G by e R f, where e, f ∈E(G), if e = f or e and f lie on a common cycle of G. Then R is an equivalence relation. 用等价关系来描述边之间的关系，到底有何用意？ Each subgraph of G induced by the edges in an equivalence class is in fact a block of G. 问题5：Block的“极大”特性和“等价类”有何关系？
间题6： 两个bock的“边界”是什么？ 1 U2 3 G:2 201 02 u3 01 V1 u W2 3 Bi B2 B3 Ba Figure 5.4:A connected graph and its blocks