## CoveringsA graph covering of a graph G is a sub-graph of G which contains either all the vertices or all the edges corresponding to some other graph. A sub-graph which contains all the vertices is called a line/edge covering. A sub-graph which contains all the edges is called a vertex covering. ## 1. Edge CoveringA set of edges which covers all the vertices of a graph G, is called a Edge covering does not exist if and only if G has an isolated vertex. Edge covering of graph G with n vertices has at least n/2 edges. ## ExampleIn the above graph, the red edges represent the edges in the edge cover of the graph. ## Minimal Line coveringA line covering M of a graph G is said to be minimal line cover Or minimal edge cover is an edge cover of graph G that is No minimal line covering contains a cycle. ## ExampleFrom the above graph, the sub-graph having edge covering are: M Here, M ## Minimum Line CoveringA Every minimum edge cover is a minimal edge cove, but the converse does not necessarily exist. The number of edges in a minimum line covering in G is called the line covering number of G and it is denoted by α ## ExampleFrom the above graph, the sub-graph having edge covering are: M In the above example, M ## 2. Vertex CoveringA set of vertices which covers all the nodes/vertices of a graph G, is called a Example In the above example, each red marked vertex is the vertex cover of graph. Here, the set of all red vertices in each graph touches every edge in the graph. ## Minimal Vertex CoveringA vertex M of graph G is said to be minimal vertex covering if no vertex can be deleted from M. ## ExampleThe sub- graphs that can be derived from the above graph are: M Here, M ## Minimum Vertex CoveringA minimal vertex covering is called when minimum number of vertices are covered in a graph G. It is also called smallest minimal vertex covering. The ## Example 1In the above graphs, the vertices in the minimum vertex covered are red.
## Example 2The sub- graphs that can be derived from the above graph are: M Here, M1 is a minimum vertex cover of G, as it has only two vertices. Therefore, α Next Topic# |