The one-way street problem Definitions The one-way street problem. ¾ Traffic-flow problem in graph theory. An orientation of a graph G is an assignment of a direction to each edge of A city has a number of ¾ Graph: locations, some of which are Vertices = locations G (obtaining as a result a digraph). joined by two way streets. Edges = two-way streets Making all streets one way ¾ Orientation of the graph. presumably will cut down on Let G be a connected graph and α one of traffic congestion. its edges. We say that α is a bridge of G Make sure that for every pair ¾ The resulting digraph should be strongly connected. of locations x and y it is if G α′ (the graph obtained by removing α possible (legally) to go from x to y and from y to x. from G ) is disconnected. Theorem (Robbins, 1939) Eulerian paths and chains A graph G has a strongly connected A graph (digraph) with multiple edges (arcs) is called a multigraph (multidigraph). orientation if and only if G has no bridges. All definitions we have for graphs and digraphs (degrees, paths, chains, connectedness, etc.) can be extended to multigraphs and multidigraphs. A chain (in a multigraph G) or a path (in a multidigraph D) is eulerian if it uses all edges of G or arcs of D exactly once. Existance of Eulerian chains Existance of Eulerian paths Theorem (Euler). A multigraph G has an eulerian Theorem (Good, 1946). A multidigraph D has an closed chain if and only if G is connected up to eulerian closed path if and only if D is weakly isolated vertices and every vertex of G has even connected up to isolated vertices and for every degree. vertex of D, indegree equals outdegree. Theorem (Euler) A multigraph G has an eulerian Theorem (Good, 1946) A multidigraph D has an chain if and only if G is connected up to isolated eulerian path if and only if D is weakly connected up vertices and the number of vertices of odd degree to isolated vertices and for all vertices, with the of G is either 0 or 2. possible exception of two, indegree equals outdegree, and for at most two vertices indegree and outdegree differ by one. 1.