freepeople性欧美熟妇, 色戒完整版无删减158分钟hd, 无码精品国产vα在线观看DVD, 丰满少妇伦精品无码专区在线观看,艾栗栗与纹身男宾馆3p50分钟,国产AV片在线观看,黑人与美女高潮,18岁女RAPPERDISSSUBS,国产手机在机看影片

正文內(nèi)容

bipartitematching(文件)

2025-08-02 17:50 上一頁面

下一頁面
 

【正文】 5. Since the graph is smaller (one fewer edge), by induction, 6. there is a perfect matching in this smaller graph, 7. hence there is a perfect matching in the original graph. Hall’s Theorem: If |N(S)| = |S| for every subset S of V, then there is a perfect matching. Proof of Hall’s Theorem Case 2: Suppose there is a proper subset S with |N(S)| = |S|. S N(S) Divide the graph into two smaller graphs G1 and G2 (so we can apply induction) Find a perfect matching in G2 by induction. Find a perfect matching in G1 by induction. G1 G2 Then we are done. Proof of Hall’s Theorem S N(S) Why there is a perfect matching in G2? To apply Hall’s, we want to show for any subset T of S, |N(T) G2| = |T|. T |N(T) G2| G2 Hall’s Theorem: If |N(S)| = |S| for every subset S of V, then there is a perfect matching. Proof of Hall’s Theorem S N(S) Why there is a perfect matching in G2? For any subset T S, N(T) is contained in G2. Hence, |N(T) G2| = |N(T)| = |T|. Therefore, by induction, there is a perfect matching in G2. T G2 Find a perfect matching in G2 by induction. |N(T) G2| Why there is a perfect matching in G1? Proof of Hall’s Theorem S N(S) T N(T) G1 For any subset T, we want to show |N(T) G1| = |T| to apply induction. 1. Consider T, by assumption, |N(T)| = |T| 2. Can we conclude that |N(T) G1| = |T|? 3. No, because N(T) may intersect N(S)! Now what? Why there is a perfect matching in G1? Proof of Hall’s Theorem S N(S) T N(T) G1 (red) |S|=|N(S)| For any subset T, we want to show |N(T) G1| = |T| to apply induction. 1. Consider S T, by assumption, |N(S T)| = |S T| (the green areas). 2. Since |S|=|N(S)|, |N(S T) – N(S)| = |S T S| (the red areas). 3. So |N(T) G1| = |N(S T) – N(S)| = |S T – S| = |T|, as required. N(S T) (green) Hall’s Theorem: If |N(S)| = |S| for every subset S of V, then there is a perfect matching. Proof of Hall’s Theorem Case 2: Suppose there is a subset S with |N(S)| = |S|. S N(S) Divide the graph into two smaller graphs G1 and G2 (so we can apply induction) Find a perfect matching in G2 by induction. Find a perfect matching in G1 by induction. G1 G2 Now we are done. Bipartite Matching and Hall’s Theorem Hall’s theorem is a fundamental theorem in graph theory. In this cours
點擊復制文檔內(nèi)容
數(shù)學相關推薦
文庫吧 www.dybbs8.com
備案圖鄂ICP備17016276號-1