Math 3336 Section 10.2 Graph terminology and Special Types of Graphs Definition: A graph is an object consisting of two sets called its vertex set and its edge set. The vertex set is a finite nonempty set. The edge set may be empty, but otherwise its elements are two-element subsets of the vertex set. Definition: The elements of the vertex set are called vertices and the elements of the edge set are called edges. Definitions: If {uu, vv} is an edge of a graph we say {uu, vv} joins or connects the vertices uu and vv, and that uu and vv are adjacent to each other. The edge {uu, vv} is incident to each uu and vv. Two edges incident to the same vertex are called adjacent edges. A vertex incident to no edges is called isolated. The set of all neighbors of a vertex vv of GG = (VV, EE), denoted by NN(vv), is called the neighborhood of vv. If AA is a subset of VV, we denote by NN(AA) the set of all vertices in GG that are adjacent to at least one vertex in AA. The degree of a vertex in a undirected graph is the number of edges incident with it, except that a loop at a vertex contributes two to the degree of that vertex. The degree of the vertex vv is denoted by dddddd(vv). Example: Page 1 of 7
Theorem (Handshaking Theorem): If GG = (VV, EE) is an undirected graph with mm edges, then 2mm = deg (vv) Proof: vv VV. Example: How many edges are there in a graph with 10 vertices of degree six? Solution: Example: If a graph has 5 vertices, can each vertex have degree 3? Solution: Page 2 of 7
Theorem: An undirected graph has an even number of vertices of odd degree. Proof: Special Types of Graphs 1. A complete graph on nn vertices, denoted by KK nn, is the graph that have all possible edges. Theorem: The number of edges in a complete graph KK nn is given by the formula vv(vv 1) ee = 2 Proof: Page 3 of 7
2. If nn is an integer greater than or equal to 3, the cyclic graph on nn vertices, denoted CC nn, is the graph having vertex set {1, 2, 3,, nn} and edge set {{1,2}, {2, 3}, {3, 4},, {nn 1, nn}}. 3. A wheel WW nn is obtained by adding an additional vertex to a cycle CC nn for nn 3 and connecting this new vertex to each of the nn vertices incc nn by new edges. 4. A simple graph GG is bipartite if VV can be partitioned into two disjoint subsets VV 1 and VV 2 such that every edge connects a vertex in VV 1 and a vertex in VV 2. In other words, there are no edges which connect two vertices in VV 1 or in VV 2. It is not hard to show that an equivalent definition of a bipartite graph is a graph where it is possible to color the vertices red or blue so that no two adjacent vertices are the same color. Example: Determine if G and H are bipartite graphs. Page 4 of 7
Example: Show that CC 6 is bipartite. Example: Show that C 3 is not bipartite. 5. A complete bipartite graph KK mm,nn is a graph that has its vertex set partitioned into two subsets VV 1 of size m and VV 2 of size n such that there is an edge from every vertex in VV 1 to every vertex in VV 2. Page 5 of 7
Bipartite graphs are used to model applications that involve matching the elements of one set to elements in another, for example: Job assignments - vertices represent the jobs and the employees, edges link employees with those jobs they have been trained to do. A common goal is to match jobs to employees so that the most jobs are done. Marriage - vertices represent the men and the women and edges link a man and a woman if they are an acceptable spouse. We may wish to find the largest number of possible marriages. HALL S MARRAGE THEOREM Suppose that there are m men and n women on an island. Each person has a list of members of the opposite gender acceptable as a spouse. We construct a bipartite graph GG(VV 1, VV 2 ), where VV 1 is the set of men and VV 2 is the set of women. There is an edge between a man and a women if they find each other acceptable as a spouse. A matching in this graph consists of a set of edges, where each pair of endpoints of an edge is a husband- wife pair. A maximum matching is a largest possible set of married couples. A complete matching of VV 1 is a set of married couples where every man is married, but possibly not all women. Page 6 of 7
HALL S MARRAGE THEOREM The bipartite graph GG = (VV, EE) with partition (VV 1, VV 2 ) has a complete matching from VV 1 to VV 2 if and only if NN(AA) AA for all subsets AA of VV 1. Page 7 of 7