VertexCoverQ[g, vlist] yields True if the vertex list vlist is a vertex cover of the graph g, and False otherwise.

VertexOutDegree[g] gives the list of vertex out-degrees for all vertices in the graph g.VertexOutDegree[g, v] gives the vertex out-degree for the vertex v.

VertexList[g] gives a list of all vertices in the graph g.

GraphDiameter[g] gives the maximum eccentricity of the vertices in the graph g.

TwoColoring[g] finds a two-coloring of graph g if g is bipartite. It returns a list of the labels 1 and 2 corresponding to the vertices.

VertexShape is an option and property for Graph and related functions that specifies the graphics used for vertices.

BipartiteMatchingAndCover[g] takes a bipartite graph g and returns a matching with maximum weight along with the dual vertex cover. If the graph is not weighted, it is ...

BipartiteMatching[g] gives the list of edges associated with a maximum matching in bipartite graph g. If the graph is edge weighted, then the function returns a matching with ...

OrientGraph[g] assigns a direction to each edge of a bridgeless, undirected graph g, so that the graph is strongly connected.

GraphicQ[s] yields True if the list of integers s is a graphic sequence, and thus represents a degree sequence of some graph.