Showing papers on "Complement graph published in 1969"

An algorithm for finding a fundamental set of cycles of a graph

Abstract: A fast method is presented for finding a fundamental set of cycles for an undirected finite graph. A spanning tree is grown and the vertices examined in turn, unexamined vertices being stored in a pushdown list to await examination. One stage in the process is to take the top element v of the pushdown list and examine it, i.e. inspect all those edges (v, z) of the graph for which z has not yet been examined. If z is already in the tree, a fundamental cycle is added; if not, the edge (v, z) is placed in the tree. There is exactly one such stage for each of the n vertices of the graph. For large n, the store required increases as n2 and the time as ng where g depends on the type of graph involved. g is bounded below by 2 and above by 3, and it is shown that both bounds are attained.In terms of storage our algorithm is similar to that of Gotlieb and Corneil and superior to that of Welch; in terms of speed it is similar to that of Welch and superior to that of Gotlieb and Corneil. Tests show our algorithm to be remarkably efficient (g = 2) on random graphs.

An Algorithm for Finding a Minimum Equivalent Graph of a Digraph

Abstract: : A directed graph or digraph can be thought of as a communication network among a set of persons, where the vertices of the graph correspond to persons and edges of the graph to directed channels of communication from one person to another. A person is said to be able to 'reach' another if he can send a message to that person. The present paper gives an algorithm for finding the maximum number of edges that can be removed from a digraph without affecting its reachability properties. (Author)

Maximally Distant Trees and Principal Partition of a Linear Graph

Abstract: A pair of trees of a graph is said to be maximally distant if the distance between these trees is maximum in the graph Necessary and sufficient conditions for maximally distant trees are presented Fundamental properties of the maximally distant trees provide us with the principal partition of a graph that is uniquely determined for the graph Some useful properties of the principal partition, together with the application to the classification of the trees, are discussed

On Ulam’s conjecture for separable graphs

Abstract: Ulam's conjecture, that every graph of order greater than two is determined up to isomorphism by its collection of maximal subgraphs, is verified for the case of separable graphs which have no pendant vertices. Partial results are then obtained for the case of graphs with pendant vertices. Unless otherwise stated, the graphs dealt with in this paper will be finite and undirected, and may have loops and multiple edges. Any definitions and notation not given below can be found in Berge [1]. A part Gι of a graph G is a subset of the vertices and edges of G. The end-vertices of edges in Gι need not themselves be in G\ If Gι is a part of G, G — G1 denotes that partial subgraph of G which is obtained by deleting G1 and all edges of G which are joined to vertices of G1. Now let S be some distinguished set of parts of a graph, and let S(X) — {X1} be the labelled set of these parts in the graph X We call two graphs G, H S-equίvalent if | S(G) \ = | S(H) | = M« oo) and, possibly after relabelling, G - Gι ^ H - if*(l ^ i ^ M). G\ Hι will be referred to as corresponding parts. In [8] Ulam proposed the following conjecture. CONJECTURE A. Vertex-equivalent graphs of order greater than two are isomorphic.

Correction to “An application of graph theory to algebra”

Abstract: PROOF. Let r' be the result of deleting e, e', and X. The theorem holds for r' by induction. Any unicursal path on r' has the form 7172 ... 7r* where each 7ri is a path starting and ending at P but not meeting P between. Clearly n is the number of edges leaving P in r, and so is the same for all paths. Let X be the path ee' from P to P in r. We get all possible unicursal paths on r by starting with such paths on rI and inserting X, getting X7w1 * * * 7r., T.1X .. *nX . . . I 7ri * * * 7nAX. Assuming that e, e' are the last two edges in the chosen ordering of the edges, we have-(71 . . . 7-jiX-i+1 * * * 7rn) =E(7ri . . . 7) Thus E(7r) = (n+1) ,E(r') =0, the first sum being over all unicursal paths on r and the second over such paths on r'. We now consider Case 2 of [1]. If P =A, we can repeat the argument of Case 2 using B and C in place of B and A with only minor modifications. This will be possible provided C#A, but if C=A, the lemma applies with X = B. Suppose now that P = B. Let U be the set of unicursal paths on r starting at A, U' the set of such paths which begin with e, and Ui the set of unicursal paths on ri starting at A. Then the argument of [1, Case 2] shows that U= U'YUUi, a disjoint union. Since the theorem holds for U by what we have just proved, and also for Us, we see that E(r') = 0 where 7r' runs over all elements of U'. But there is a one-to-one correspondence between U' and the set of unicursal paths starting from B, given by ee'e, ... e. elel *.*.* ene. Since n =E-2 is even, e(ee'e1 . . . en) = -e(e' * * * e,e). Therefore the theorem holds in this case also. Finally, we consider Case 3. If there is an edge not meeting P, choose it for e4. Then P A, B and we are done. Suppose every edge meets P. Let P, A1, * * * I A. be the vertices. Then E=2V=2n+2.

Bounds for Maxium Parallelism in a Bilogic Graph Model of Computations

Abstract: Given an acyclic directed graph where vertices represent computational tasks, arcs represent transfer of control, and two labels—called input and output logics—associated with each vertex show either the concurrency or the mutual exclusiveness of tasks, procedures are given to determine a lower and an upper bound on the number of processors required for maximum parallelism. The lower bound is obtained via a mean path length approach, while the upper bound is based on the structure of the graph. A detailed algorithm is given for the latter. First, some reduction rules are applied yielding a subset of the vertices which can be performed in parallel. Then the maximum cut in the graph is determined taking into account mutually exclusive vertices. Results are given for example graphs.

Enumeration of all trees of a graph

Abstract: The nonsingular submatrices of the characteristic part of the fundamental cutset matrix relative to a tree uniquely correspond to the other trees of a graph G. Using this principle, a method of listing, without duplication, all trees of G is presented. For implementation of the method on a computer, a large memory is not required.

Uniqueness of the graph of a rank three group

Abstract: Let G be a transitive permutation group on a set Ω. Let {a} be an element of Ω and Δ(a) an orbit of the stabilizer Ga of {a} in G. A graph on the points of Ω can be constructed by joining {a} to Δ(a) and joining each image of {a} by an element of G to the corresponding image of Δ(a). The permutation group G is said to be of rank 3 if Ga has three orbits including {a}. In this paper, conditions are discussed which ensure that the graph of a rank three permutation group is unique in the following sense. Suppose G is a rank three permutation group and Ga has orbits {a}, Δ(a), and Γ{a). Is the graph uniquely determined once the permutation representation of Ga on Δ(a) and on Γ(a) is known ? Conditions involving orbit lengths of Ga acting on Δ(a) and on Γ(a) are found which ensure the graph is uniquely determined. The conditions specify how the graph is to be drawn. They give, therefore, a prescription for constructing rank three groups. They shed no light however on the difficult problem of the transitivity of the group of automorphisms of the graph. A few applications are given at the end to illustrate the method. They are not intended to be exhaustive. The same ideas can be used to show the nonexistence of certain rank three extensions.

On k-Stable Feedback Shift Registers

Abstract: A state graph (SG) is a directed graph with exactly one arc issuing from every vertex of the graph. The degree of an SG is the smallest integer d such that at most d, arcs are entering any vertex of the graph. An SG is said to be k-stable if it contains k≤1 cycles of unit length (loops) and these are the only cycles of the graph. A k-stable SG with mn vertices and degree d is called a (k, m, n)-SG if m≥k, d and if the distance from any vertex to a loop of the graph is at most n.

Can the Complete Graph with 2n+1 Vertices be Packed with Copies of an Arbitrary Tree having n Edges?

Abstract: (1969). Can the Complete Graph with 2n+1 Vertices be Packed with Copies of an Arbitrary Tree having n Edges? The American Mathematical Monthly: Vol. 76, No. 10, pp. 1128-1130.

Signal-flow-graph gain with respect to the general node of a graph

Abstract: A new formula for the graph gain is given, which is more general than Mason's formula and enables the graph gain to be solved with respect to the general node of the graph that may also be the loop node.

Enumeration of all cutsets of a graph

Abstract: A computer program has been written which can be used for listing all the cutsets of a graph. The program can be applied to connected graphs having a maximum of 10 vertices and 20 edges.

Density of one graph along another

Abstract: Introduction. The word "graph" shall mean the graph of a real function, and the X-projection of a graph F is the set of all abscissas of points of F. c denotes the cardinality of the continuum. If H is a collection of sets, H* denotes the union of all the sets in H. DEFINITION. Suppose F and G are graphs with X-projection [0, 1]. The statement that F is dense (c-dense) along G means that if [a, b] is a subinterval of [0, 1], then there is a point (are c-many points) of intersection of F and G with abscissa in [a, b]. In this paper, the following three theorems will be proved:

On dual of a graph on a surface of genus 1

Abstract: : The dual of a graph on a surface of a certain genus is defined. A necessary and sufficient condition for a graph to be drawn on a surface of genus 1 is derived by using the concept.

Conversion of an arbitrary node into a source node in a flow graph

Abstract: A simple procedure which converts an arbitrary node into a source node for both the Coates flow graph and Mason's signal-flow graph is presented. The technique is believed to be simpler than the well known path-inversion process.

A physical interpretation of the multiple-node removal algorithm of a flow graph

Abstract: A simple physical interpretation of the multiple-node removal algorithm for both Coates' flow graph and Mason's signal-flow graph is presented. It is shown that the transmittances associated with the branches of a reduced flow graph can be interpreted as the graph transmissions of certain subgraphs.

New analysis tool for graphs

Abstract: The variable incidence set of a graph is defined and its usefulness demonstrated in reducing a graph by identifying specified vertices and removing self loops, testing for connectedness, and generating specified segs, cosegs and cutsets.