Day 07Part 06- Examples based on Bipartite graph and Complete bipartite graph.<\/p>\n
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A Bipartite Graph<\/strong> is a graph whose vertices can be divided into two disjoint sets<\/strong> such that no two vertices within the same set are adjacent<\/strong>. A Complete Bipartite Graph<\/strong> is a special type of bipartite graph where every vertex from one set is connected to every vertex in the other set<\/strong>.<\/p>\n Consider a graph with vertices V = {A, B, C, D}<\/strong> and edges E = {(A, C), (A, D), (B, C), (B, D)}<\/strong>.<\/p>\n Divide the vertices into two sets:<\/p>\n Set 1:<\/strong> {A, B}<\/p>\n<\/li>\n Set 2:<\/strong> {C, D}<\/p>\n<\/li>\n<\/ul>\n<\/li>\n Each edge connects a vertex in Set 1<\/strong> to a vertex in Set 2<\/strong> (i.e., no two vertices in the same set are adjacent).<\/p>\n<\/li>\n Since this condition is satisfied, the graph is bipartite<\/strong>.<\/p>\n<\/li>\n<\/ol>\n A Complete Bipartite Graph K\u2098,\u2099<\/strong> has two sets of vertices with m<\/strong> and n<\/strong> vertices, where every vertex in one set is connected to all vertices in the other set<\/strong>.<\/p>\n Consider K\u2083,\u2084<\/strong>, with two sets:<\/p>\n Set X:<\/strong> {A, B, C} (3 vertices)<\/p>\n<\/li>\n Set Y:<\/strong> {D, E, F, G} (4 vertices)<\/p>\n<\/li>\n<\/ul>\n Each vertex from Set X<\/strong> connects to all vertices in Set Y<\/strong>:<\/p>\n A \u2192 {D, E, F, G}<\/p>\n<\/li>\n B \u2192 {D, E, F, G}<\/p>\n<\/li>\n C \u2192 {D, E, F, G}<\/p>\n<\/li>\n<\/ul>\n Since every vertex from one set is connected to all vertices of the other set<\/strong>, this is a Complete Bipartite Graph<\/strong>.<\/p>\n A graph is bipartite if it can be colored using two colors without any adjacent nodes sharing the same color.<\/strong> Would you like more practice problems or algorithmic methods<\/strong> to check if a graph is bipartite?<\/p>\n Here’s a clear and simplified explanation for Day 07 Part 06<\/strong> of Discrete Mathematics for Computer Engineering<\/strong>: A Bipartite Graph<\/strong> is a graph where:<\/p>\n The set of vertices can be divided into two disjoint sets<\/strong>, say U<\/strong> and V<\/strong>,<\/p>\n<\/li>\n Every edge connects a vertex from U<\/strong> to a vertex from V<\/strong>,<\/p>\n<\/li>\n There are no edges within the same set<\/strong>.<\/p>\n<\/li>\n<\/ul>\n A graph G(V, E)<\/strong> is bipartite<\/strong> if you can color all vertices using 2 colors<\/strong> such that no two adjacent vertices<\/strong> share the same color.<\/p>\n Let<\/p>\n U={u1,u2}U = \\{u_1, u_2\\}<\/span>U<\/span>=<\/span><\/span>{<\/span>u<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>,<\/span>u<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n V={v1,v2}V = \\{v_1, v_2\\}<\/span>V<\/span>=<\/span><\/span>{<\/span>v<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>,<\/span>v<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n Edges: {(u1,v1),(u2,v2)}\\{(u_1,v_1), (u_2,v_2)\\}<\/span>{(<\/span>u<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>,<\/span>v<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span>(<\/span>u<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>,<\/span>v<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)}<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n \ud83d\udccc Here, no edge connects u1u_1<\/span>u<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to u2u_2<\/span>u<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> or v1v_1<\/span>v<\/span>\u00a0Example 1: Checking If a Graph is Bipartite<\/strong><\/h3>\n
Graph:<\/strong><\/h3>\n
Solution:<\/strong><\/h3>\n
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\u00a0Example 2: Complete Bipartite Graph (K\u2083,\u2084)<\/strong><\/h3>\n
Graph:<\/strong><\/h3>\n
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Edges:<\/strong><\/h3>\n
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\u00a0Key Points:<\/strong><\/h3>\n
A complete bipartite graph K\u2098,\u2099 connects every vertex in one set to every vertex in the other set.<\/strong><\/p>\nDay 07Part 06- Examples based on Bipartite graph and Complete bipartite graph.<\/a><\/h3>\n
COMBINATORIAL ANALYSIS Lecture 29: Bipartite Graphs …<\/a><\/h3>\n
Complete Bipartite Graphs and Their Line Graphs<\/a><\/h3>\n
Bipartite Graphs and Problem Solving<\/a><\/h3>\n
Lecture 3 Bipartite Graphs<\/a><\/h3>\n
Examples based on Bipartite Graph and Complete Bipartite Graph<\/strong> \u2014 ideal for GATE, B.Tech, or competitive exam prep.<\/p>\n
\n\ud83c\udfaf What is a Bipartite Graph?<\/strong><\/h2>\n
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\n\u2705 Definition<\/strong>:<\/h3>\n
\n\ud83e\uddea Example 1: Simple Bipartite Graph<\/strong><\/h2>\n
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