with vertices and edges is - an euler tour of directed graph exactly once according to its direction cycle that traverses each edge of such tour always exists if is connected and the in-degree equals the describe an ( )-time algorithm for out-degree of each vertex in finding an euler tour of such directed graph - company named rt& has network of switching stations connected by high-speed communication links each customer' phone is directly connected to one station in his or her area the engineers of rt& have developed prototype video-phone system that allows two customers to see each other during phone call in order to have acceptable image qualityhoweverthe number of links used to transmit video signals between the two parties cannot exceed suppose that rt& ' network is represented by graph design an efficient algorithm that computesfor each stationthe set of stations it can reach using no more than links - the time delay of long-distance call can be determined by multiplying small fixed constant by the number of communication links on the telephone network between the caller and callee suppose the telephone network of company named rt& is tree the engineers of rt& want to compute the maximum possible time delay that may be experienced in long-distance call given tree the diameter of is the length of longest path between two nodes of give an efficient algorithm for computing the diameter of - tamarindo university and many other schools worldwide are doing joint project on multimedia computer network is built to connect these schools using communication links that form tree the schools decide to install file server at one of the schools to share data among all the schools since the transmission time on link is dominated by the link setup and synchronizationthe cost of data transfer is proportional to the number of links used henceit is desirable to choose "centrallocation for the file server given tree and node of the eccentricity of is the length of longest path from to any other node of node of with minimum eccentricity is called center of design an efficient algorithm thatgiven an -node tree computes center of is the center uniqueif nothow many distinct centers can tree haveis compact if there is some - say that an -vertex directed acyclic graph way of numbering the vertices of with the integers from to such contains the edge (ijif and only if jfor all ij in [ that is compact give an ( )-time algorithm for detecting if