Basic Operations on Graphs Lecture 5 Basic Operations
Basic Operations on Graphs Lecture 5.
Basic Operations on Graphs • • • Deletion of edges Deletion of vertices Addition of edges Union Complement Join
Deletion of Edges • If G = (V, E) is a graph and e 2 E one of tis edges, then G - e : = (V, E – {e}) is a subgraph of G. In such a case we say that Ge is obtained from G by deletion of edge e.
Deletion of Vertices • Let x 2 V(G) be a vertex of graph G, then G - x is the subgraph obtained from G by removal of x grom V(G) and removal of all edges from E(G) having x as an endpoint. G – x is obtained from G by deletion of vertex x.
Exercises 01 • N 1. Show that for any set F µ E(G) the graph G-F is well-defined. • N 2. Show that for any set X µ V(G) the graph G-X is well-defined. • N 3. Show that for any set X µ V(G) and any set F µ E(G) the graph G-X-F is well-defined. • N 4. Prove that H is a subgraph of G if and only if H is obtained from G by a succession of vertex and edge deletion.
Edge Addition • Let G be a graph and (u, v) a pair of nonadjacent vertices. Let e = uv denot the new edge between u and v. By G’ = G + uv = G + e we denote the graph obtained from G by addition of edge e. In other words: • V(G’) : = V(G), • E(G’) : = E(G) [ {e}.
Graph Union Revisited • If G and H are graphs we denote by G t H their disjoint union. • Instead of G t G we write 2 G. • Generalization to n. G, for an arbitrary positive integer n: – 0 G : = ; . – (n+1)G : = n. G t G • Example: • Graph in top row C 6 t K 9 • Graph in bottom row 2 K 3.
Graph Complement • Graph complement Gc of simple graph G has V(Gc) : = V(G), but two vertices u in v are adjacent in Gc if and only if they are not adjacent in G. • For instance C 4 c is isomorphic to 2 K 2.
Graph Difference • If H is a spanning subgraph of G we may define graph difference G H as follows: • V(GH) : = V(G). • E(GH) : = E(G)E(H). G H GH
Bipartite Complement X Xb • For a bipartite graph X (with a given biparitition) one can define a bipartite complement Xb. This is the graph difference of Km, n and X: Xb = Km, n X.
Empty Graph Revisited. • The word “empty graph” is used in two meanings. • First Meaning: ; . No vertices, no edges. • Second Meaning: En : = Knc. = n. K 1. There are n vertices, no edges. • E 0 = ; = 0. G will be called the void graph or zero graph.
Graph Join • Join of graphs G and H is denoted by G*H and defined as follows: • G*H : = (Gc t Hc )c • In particular, this means that Km, n is a join of two empty graphs En and Em.
- Slides: 12