Showing papers on "Adjacency list published in 1969"

Distinguishability Criteria in Oriented Graphs and Their Application to Computer Diagnosis-I

Abstract: Discrete sequential systems like the functional elements of a digital computer can be represented by directed graphs. In this paper we study an application of graph theory to computer diagnosis. Specifically, we develop the distinguishability criteria in directed graphs and derive bounds on the number of test points needed to locate faults in a sequential system.

The cube of every connected graph is 1-hamiltonian

Abstract: Let G be any connected graph on 4 or more points. The graph G3 has as it s point set that of C, and two di s tinct pointE U and v are adjacent in G3 if and only if the distance betwee n u and v in G is at most three. It is shown that not only is G" hamiltonian, but the removal of any point from G" still yields a hamiltonian graph. Let G be a graph (finite, undirected , with no loops or multiple lines). A waLk of G is a finit e alternating seque nce of points and lines of G, beginning and ending with a point and where each lin e is incide nt with the points immediately preceding and followin g it. A walk in whi c h no point is repeated is called a path; the Length of a path is the number of lines in it. A graph G is connected if between e very pair of di stin ct points th er e exis ts a path, and for s uc h a graph , the distance between two points u and v is defin e d as the le ngth of the shortest path if u 0;1= v and zero if u = v. A walk with at least three points in whic h the fir st and last points are the same but all other points are di s tinc t i s called a cycle. A cycle containing all points of a graph G is called a hamiLtonian cycle of G, and G itself a hamiLtonian graph. Throughout the literature of graph theory there have been defin ed many gr a ph-valued func ­ tions / o n the class of graphs. In certain ins ta nces r es ults have been obtained to show tha t if G is connecte d a nd has s uffi ciently many points, then the graph/(G) (or its ite rates/n(G» is a hamiltoni an graph. Examples of suc h include th e line-graph func ti on L(G) and the total graph function T(G) (see [2 , 1] ,1 respectively). The Line-graph L (G) of graph G is a graph whose poi nt set can b e put in one-to-one correspond­ ence with the lin e set of G such that adjacency is preserved. The totaL graph T(G} has its point set in one-to-one correspondence with the set of points and lines of G in such a way that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent or incident. Another example which always yields a hamiltonian graph is the c ube function. In fact, if x is any line in a connected graph G with at least three points, the n the c ube of G has a h amiltonian cycle containing x. This follows from a result due to Karaganis [4] by whic h the cube of any con­ nected graph G on p( ;?: 3) points turns out to be hami ltonian-connected , i.e., between any two points the r e exists a path containing all points of G. Now if x is any line joining points u and v in G, then the addition of x to the hamiltonian path between u and v in the c ube of the graph produces a hamil­

On the Tours of a Traveling Salesman

Abstract: Adjacency properties of tours on their convex hull are discussed. A rule is given by which it can be tested whether any two tours are adjacent vertices on this convex hull or not. Based on this rule an algorithm is described for generating all the adjacent tours of a given tour.