Graph Theory Introducton

Graph Theory Vertex: A point. An intersection of two lines (edges). T. Serino Vertex: A point. An intersection of two lines (edges). Edge: A line (or curve) connecting two vertices. Loop: An edge that connects a vertex to itself only.

Graph Theory T. Serino Ex) Represent the "Konigsberg Bridge“ problem using a vertex-edge graph. A B C D A B C D * Vertices represent locations. * Edges represent “connections” between those locations.

Graph Theory Ex) Represent this map using a vertex-edge graph. W K C U T. Serino Ex) Represent this map using a vertex-edge graph. Hint: On map problems, place vertices relative to their actual locations on the map. W K C U O N * Edges represent borders in a map problem.

Graph Theory Ex) Represent a floor plan using a vertex-edge graph. C F T. Serino Ex) Represent a floor plan using a vertex-edge graph. C F H J M P Outside O * Rooms are locations (vertices). * Doorways are connections between locations (edges).

The loop counts twice because it touches the vertex twice. Graph Theory T. Serino The degree of a vertex is the number of edges "entering" the vertex. Vertex A has Degree 2 1 2 The loop counts twice because it touches the vertex twice. Vertex C has Degree 3 2 3 Vertex D has Degree 4 2 3 1 1 4

Graph Theory Odd and Even vertices T. Serino Odd and Even vertices If the degree of a vertex is an odd number, then the vertex is considered an odd vertex. If the degree of the vertex is an even number, then the vertex is considered an even vertex.

Graph Theory 2 odd vertices T. Serino Ex) Identify the degree of each vertex and determine how many odd vertices are contained in the following graph. Degree 4 Degree 4 Degree 2 odd vertices F Degree 1 Degree 4 Degree 3

Graph Theory T. Serino A path is a sequence of adjacent vertices and the edges connecting them. Given the graph to the left, some examples of paths could be: ABC BCDD CDBCA

Graph Theory T. Serino A circuit is a path that begins and ends at the same vertex. Given the graph to the left, some examples of circuits could be: ABCA CDDBAC

Graph Theory On a connected graph, you can draw a path T. Serino On a connected graph, you can draw a path from one vertex to any other vertex. The following are all connected graphs.

Graph Theory If a graph is not connected, it is disconnected. T. Serino If a graph is not connected, it is disconnected. The following are all disconnected graphs. Could you draw a bridge (an edge) that could make each of these graphs connected?

Graph Theory T. Serino A bridge is an edge that if removed from a connected graph would create a disconnected graph.

athematical M D ecision aking