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In this set of notes, we focus on the case when the underlying graph is bipartite. We start by introducing some basic graph terminology. A graph G= (V;E) consists of a set V of vertices and a set Eof pairs of vertices called edges. For an edge e= (u;v), we say that the endpoints of eare uand v; we also say that eis incident to uand v. A graph
Bipartite graphs. set of pairwise adjacent vertices in a graph is called a clique. A set of pairwise non-adjacent vertices in a graph is called an independent set. graph G is bipartite if V (G) is the union of two inde-pendent sets of G. If these are disjoint, they are called the partite sets of G. Examples.
A graph. G = (V; E) is bipartite if the vertex set V can be partitioned into two sets A and B (the bipartition) such that no edge in E has both endpoints in the same set of the bipartition. A matching M E is a collection of edges such that every vertex of V is incident to at most one edge of M.
α′(G) Proof. Let G = (X, Y ; E) be a bipartite graph with parts X and . By Observation C, we need only to prove α′(G) ≥ β(G). be a vertex cover of G with |Q| = β(G). Claim: (i) ∀A ⊆ Q ∩ X, |N(A) − Q ∩ Y | ≥ |A|. ⊆ Q ∩ Y , |N(B) − Q ∩ X| ≥ |B|. Proof of Claim (i). If for some A ⊆ Q ∩ X |N(A) − Q ∩ Y | < |A|,
5 lis 2011 · We shall call a graph with no multiple edges simple. If G is simple and e ∈ E (G) an edge with ψ G ( e) = uv, we shall write e = uv. The degree dG(v) of a vertex v is the number of edges incident with v. We denote the minimum and maximum degrees of G by δ ( G) and Δ ( G ), respectively.
demonstrate how to use bipartite graphs to solve problems. 1 Graphs A Graph G is defined to be an ordered triple (V(G),E(G),φ(G)), where V(G) is the nonempty set of vertices of G, E(G) is the set of edges of G, and φ(G) associates to each edge in E(G) two unordered vertices in V(G). If φ(e) = uv, for e ∈ E(G) and v,u ∈ V(G), then we say ...
A graph G = (V;E) is called bipartite if there is a partition of V into two disjoint subsets: V = L[R, such every edge e 2E joins some vertex in L to some vertex in R. When the bipartition V = L [R is speci ed, we sometimes denote this bipartite graph as G = (L;R;E). Theorem 2. G = (V;E) is bipartite if and only if G has no cycles of odd length ...
