Null graph

In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph"). The order-zero graph, K_0, is the unique graph having no vertices (hence its order is zero). It follows that K_0 also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude K_0 from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including K_0 as a valid graph is useful depends on context. On the positive side, K_0 follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair (V, E) for which the vertex and edge sets, V and E, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures K_0 is useful for defining the base case for recursion (by treating the null tree as the child of missing edges in any non-null binary tree, every non-null binary tree has exactly two children). On the negative side, including K_0 as a graph requires that many well-defined formulas for graph properties include exceptions for it (for example, either "counting all strongly connected components of a graph" becomes "counting all non-null strongly connected components of a graph", or the definition of connected graphs has to be modified not to include K_0). To avoid the need for such exceptions, it is often assumed in literature that the term graph implies "graph with at least one vertex" unless context suggests otherwise. In , the order-zero graph is, according to some definitions of "category of graphs," the initial object in the category. K_0 does fulfill (vacuously) most of the same basic graph properties as does K_1 (the graph with one vertex and no edges). As some examples, K_0 is of size zero, it is equal to its complement graph _0, a forest, and a planar graph. It may be considered undirected, directed, or even both; when considered as directed, it is a directed acyclic graph.