CoveringsA graph covering of a graph G is a subgraph of G which contains either all the vertices or all the edges corresponding to some other graph. A subgraph which contains all the vertices is called a line/edge covering. A subgraph 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 line cover or edge cover of G. 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 if no edge can be deleted from M. Or minimal edge cover is an edge cover of graph G that is not a proper subset of any other edge cover. No minimal line covering contains a cycle. ExampleFrom the above graph, the subgraph having edge covering are: M_{1} = {{a, b}, {c, d}} M_{2} = {{a, d}, {b, c}} M_{3} = {{a, b}, {b, c}, {b, d}} M_{4} = {{a, b}, {b, c}, {c, d}} Here, M_{1}, M_{2}, M_{3} are minimal line coverings, but M_{4} is not because we can delete {b, c}. Minimum Line CoveringA minimal line covering with minimum number of edges is called a minimum line covering of graph G. It is also called smallest minimal line covering. 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_{1}. ExampleFrom the above graph, the subgraph having edge covering are: M_{1} = {{a, b}, {c, d}} M_{2} = {{a, d}, {b, c}} M_{3} = {{a, b}, {b, c}, {b, d}} M_{4} = {{a, b}, {b, c}, {c, d}} In the above example, M_{1} and M_{2} are the minimum edge covering of G and α_{1} = 2. 2. Vertex CoveringA set of vertices which covers all the nodes/vertices of a graph G, is called a vertex cover for G. 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_{1} = {b, c} M_{2} = {a, b, c} M_{3} = {b, c, d} Here, M_{1} and M_{2} are minimal vertex coverings, but in M_{3} vertex 'd' can be deleted. 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 number of vertices in a minimum vertex covering in a graph G is called the vertex covering number of G and it is denoted by α_{2}. Example 1In the above graphs, the vertices in the minimum vertex covered are red. α_{2} = 3 for first graph. And α_{2} = 4 for the second graph. Example 2The sub graphs that can be derived from the above graph are: M_{1} = {b, c} M_{2} = {a, b, c} M_{3} = {b, c, d} Here, M1 is a minimum vertex cover of G, as it has only two vertices. Therefore, α_{2} = 2.
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