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In graph theory, a mixed graph G = (V, E, A) is a graph consisting of a set of vertices V, a set of (undirected) edges E, and a set of directed edges (or arcs) A. [ 1 ] Definitions and notation
Route inspection problem (also called Chinese postman problem) for mixed graphs (having both directed and undirected edges). The program is solvable in polynomial time if the graph has all undirected or all directed edges. Variants include the rural postman problem. [3]: ND25, ND27 Clique cover problem [2] [3]: GT17
In mathematics, the capacitated arc routing problem (CARP) is that of finding the shortest tour with a minimum graph/travel distance of a mixed graph with undirected edges and directed arcs given capacity constraints for objects that move along the graph that represent snow-plowers, street sweeping machines, or winter gritters, or other real-world objects with capacity constraints.
The mixed Chinese postman problem (MCPP or MCP) is the search for the shortest traversal of a graph with a set of vertices V, a set of undirected edges E with positive rational weights, and a set of directed arcs A with positive rational weights that covers each edge or arc at least once at minimal cost. [1]
The first textbook on graph theory was written by Dénes Kőnig, and published in 1936. [26] Another book by Frank Harary , published in 1969, was "considered the world over to be the definitive textbook on the subject", [ 27 ] and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other.
An extension of Robbins' theorem to mixed graphs by Boesch & Tindell (1980) shows that, if G is a graph in which some edges may be directed and others undirected, and G contains a path respecting the edge orientations from every vertex to every other vertex, then any undirected edge of G that is not a bridge may be made directed without changing the connectivity of G.
Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existence of graphs for which both the girth and the number of colors needed are ...
Ancestral graphs are mixed graphs used with three kinds of edges: directed edges, drawn as an arrow from one vertex to another, bidirected edges, which have an arrowhead at both ends, and undirected edges, which have no arrowheads. It is required to satisfy some additional constraints: