liteproof.blogg.se - Discrete mathematics with graph theory 3rd edition solutions with tips

It follows that a directed graph is an oriented graph if and only if it hasn't any 2-cycle. at most one of ( x, y) and ( y, x) may be arrows of the graph). Oriented graphs are directed graphs having no bidirected edges (i.e.There are extensions of quasi-transitive digraphs called k-quasi-transitive digraphs. A semicomplete digraph is a quasi-transitive digraph. Note that there can be just one arc between x and z or two arcs in opposite directions. Quasi-transitive digraphs are simple digraphs where for every triple x, y, z of distinct vertices with arcs from x to y and from y to z, there is an arc between x and z.Every semicomplete digraph is a semicomplete multipartite digraph, where the number of vertices equals the number of partite sets. Semicomplete digraphs are simple digraphs where there is an arc between each pair of vertices.Note that there can be one arc between x and y or two arcs in the opposite directions. Semicomplete multipartite digraphs are simple digraphs in which the vertex set is partition into partite sets such that for every pair of vertices x and y in different partite sets, there is an arc between x and y.It follows that a complete digraph is symmetric. Complete directed graphs are simple directed graphs where each pair of vertices is joined by a symmetric pair of directed arcs (it is equivalent to an undirected complete graph with the edges replaced by pairs of inverse arcs).Some authors describe digraphs with loops as loop-digraphs. As already introduced, in case of multiple arrows the entity is usually addressed as directed multigraph. Simple directed graphs are directed graphs that have no loops (arrows that directly connect vertices to themselves) and no multiple arrows with same source and target nodes.

Symmetric directed graphs are directed graphs where all edges are bidirected (that is, for every arrow that belongs to the digraph, the corresponding inversed arrow also belongs to it).

More specifically, directed graphs without loops are addressed as simple directed graphs, while directed graphs with loops are addressed as loop-digraphs (see section Types of directed graphs). On the other hand, the aforementioned definition allows a directed graph to have loops (that is, arcs that directly connect nodes with themselves), but some authors consider a narrower definition that doesn't allow directed graphs to have loops. More specifically, these entities are addressed as directed multigraphs (or multidigraphs). The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a multiset). It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, links or lines.

A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.

V is a set whose elements are called vertices, nodes, or points.

In formal terms, a directed graph is an ordered pair G = ( V, A) where

2.2 Digraphs with supplementary properties.