Showing papers on "Path graph published in 1993"
TL;DR: The problem of increasing the connectivity1 of a graph at an optimal cost is studied, and an efficient approximation schemes that come within a constant factor from the optimal are focused on.
122 citations
TL;DR: It is proved that for each p ≥ 1, there is a constant c (depending only on p) such that the Ramsey number r ( G, G ) ≤ cn for eachp -arrangeable graph G of order n.
113 citations
TL;DR: It is proved that every triangle-free graph on n \geq 4 vertices has at most $2 n /2 $ or $5 \cdot 2^{( n - 5 )/2} $ independent sets maximal under inclusion, whether n is even or odd.
Abstract: In this paper, it is proved that every triangle-free graph on $n \geq 4$ vertices has at most $2^{n /2} $ or $5 \cdot 2^{( n - 5 )/2} $ independent sets maximal under inclusion, whether n is even or odd In each case, the extremal graph is unique If the graph is a forest of odd order, then the upper bound can be improved to $2^{( n - 1 )/2} $
105 citations
TL;DR: It is proved that, for every positive integer k, there is an integer N such that every 3-connected graph with at least N vertices has a minor isomorphic to the k -spoke wheel or K 3, k ; and that every internally 4-connected graphs has aMinor isomorphism to the 2 k-spoke double wheel, the k-rung circular ladder, thek -rung Mobius ladder, or K 4, k.
102 citations
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TL;DR: In this article, it was shown that the Coxeter graph and Sylvester graph are induced subgraphs, and that the Sylvesters and Coxeter graphs can be viewed as a set of connected sub-graphs.
79 citations
TL;DR: This paper finds exact values of PR ( k, l ) for all k and l and proves that every triangle-free planar graph on n vertices contains an independent set of size ⌊ n /3⌋ + 1.
53 citations
TL;DR: It is proved that the stability number of bull-free chair-free graphs can be found in polynomial time.
43 citations
30 Sep 1993
TL;DR: A new approach to the problem of finding edge-disjoint paths in a planar, undirected graph with “classical” case where an instance must additionally fulfill the so-called evenness-condition is introduced.
Abstract: In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph s.t. each path connects two specified vertices on the outer face boundary. We will focus on the “classical” case where an instance must additionally fulfill the so-called evenness-condition. The fastest algorithm for this problem known from the literature requires \(\mathcal{O}\left( {n^{{5 \mathord{\left/{\vphantom {5 3}} \right.\kern-
ulldelimiterspace} 3}} \left( {\log \log n} \right)^{{1 \mathord{\left/{\vphantom {1 3}} \right.\kern-
ulldelimiterspace} 3}} } \right)\) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which yields an \(\mathcal{O}\left( n \right)\) algorithm.
34 citations
TL;DR: In this paper, properties of graphs satisfying the adjacency property are presented and it is shown that all sufficiently large Paley graphs satisfy P(m,n,k).
Abstract: In the application of graph theory to problems arising in network design, the requirements of the network can be expressed in terms of restrictions on the values of certain graph parameters such as connectivity, edge-connectivity, diameter, and independence number. In this paper, we focus on networks whose requirements translate into adjacency restrictions on the graph representing the network. More specifically, a graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices there are at least k other vertices, each of which is adjacent to the first m vertices but not adjacent to any of the latter n vertices. The problem that arises is that of characterizing graphs having property P(m,n,k). In this paper, we present properties of graphs satisfying the adjacency property. In particular, for q 1(mod 4), a prime power, the Paley graph Gq of order q is the graph whose vertices are elements of the finite field q; two vertices are adjacent if and only if their difference is a quadratic residue. For any m, n, and k, we show that all sufficiently large Paley graphs satisfy P(m,n,k). © 1993 by John Wiley & Sons, Inc.
33 citations
TL;DR: It is feasible to reason without approximation about completely specified paths through a complete directed acyclic graph, as opposed to an incompletely specified path that may also include other vertices than those stated.
28 citations
TL;DR: A slight modification of the algorithm can be used for computing the average distance of a tree with weighted edges in time O(n), where n is the number of vertices.
TL;DR: It is shown that a maximal triangle-free graph on n vertices with minimum degree δ contains an independent set of 3δ − n Vertices which have identical neighborhoods, which yields a simple proof that if the binding number of a graph is at least 3/2 then it has a triangle.
Abstract: We show that a maximal triangle-free graph on n vertices with minimum degree δ contains an independent set of 3δ − n vertices which have identical neighborhoods. This yields a simple proof that if the binding number of a graph is at least 3/2 then it has a triangle. This was conjectured originally by Woodall. © 1993 John Wiley & Sons, Inc.
TL;DR: Imagine the edges of the complete graph on the natural numbers are colored with 2 colors and there is a monochromatic infinite path such that, for infinitely many n, the set{1,..., n} contains at least the first .21n vertices of the path.
16 Jun 1993
TL;DR: In this paper, the authors examined how the number of linear intervals needed to obtain shortest path routings in networks is affected by the product, join and composition operations on graphs, and established the precise value of the minimum number of intervals required to achieve shortest path routing in the network considered in [11].
Abstract: We consider linear interval routing schemes studied in [3, 5] from a graph theoretic perspective. We examine how the number of linear intervals needed to obtain shortest path routings in networks is affected by the product, join and composition operations on graphs. This approach allows us to generalize some of the results in [3, 5] concerning the minimum number of intervals needed to achieve shortest path routings in certain special classes of networks. We also establish the precise value of the minimum number of intervals needed to achieve shortest path routings in the network considered in [11].
TL;DR: The efficiency of a graph G =( V, E ) equals the maximum efficiency of any subset S of vertices of V and an NP-completeness proof is given for the problem of deciding if an arbitrary planar bipartite graph has a set S such that e ( S )≥ k for some positive integer k .
03 May 1993
TL;DR: The 3-edge-connectivity augmentation problem for a specified set of vertices, where the graph can have multiple edges, is addressed and approximation algorithms are given.
Abstract: The 3-edge-connectivity augmentation problem for a specified set of vertices, where the graph can have multiple edges, is addressed. Both the weighted version, in which there may exist some distinct edge costs, and the unweighted version are treated. Approximation algorithms are given. >
TL;DR: It is shown that the recognition problem associated with the group path problem is NP-complete in general, and an O(| G |·|E|+|V|) time algorithm for the grouppath problem on a chordal graph is presented.
TL;DR: This simple generalization of the classic Erdős-Renyi random graph process leads to some challenging mathematical problems and is a process related to a variety of physical applications.
Abstract: Starting with n vertices and no edges, sequentially introduce edges so as to obtain a sequence of graphs each having no vertex of degree greater than f The latter are called f-graphs At each step the edge to be added is selected with equal probability from among those edges whose addition would not violate the f-degree restriction A terminal graph of this procedure is called a sequentially generated random edge maximal f-graph and the procedure the random f-graph process of order n This simple generalization of the classic Erdős-Renyi random graph process leads to some challenging mathematical problems and is a process related to a variety of physical applications
TL;DR: A graph with n vertices that contains no triangle and no 5-cycle and minimum degree exceeding n/4 contains an independent set with at least (3n)/7 vertices, which is best possible.
Abstract: A graph with n vertices that contains no triangle and no 5-cycle and minimum degree exceeding n/4 contains an independent set with at least (3n)/7 vertices. This is best possible. The proof proceeds by producing a homomorphism to the 7-cycle and invoking the No Homomorphism Lemma. For k ≥ 4, a graph with n vertices, odd girth 2k+1, and minimum degree exceeding n/(k+1) contains an independent set with at least kn/(2k+1) vertices; however, we suspect this is not best possible. © 1993 John Wiley & Sons, Inc.
TL;DR: It is proved that if a graph satisfies the conditions above except that it has too many vertices of degree 2, then the conclusion fails, related to a conjecture of Benhocine et al.
TL;DR: A new upper bound of 36 vertices for the least order of such cubic graphs is established and is realized by cubic graphs without cut-vertices.
Abstract: The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all possible paths in G. The entry (i,j) of this matrix is the number of paths in G having initial vertex i and length j. It is known that there are cubic graphs on 62 vertices having the same path layer matrix (A. A. Dobrynin. J Graph Theory 17 (1993) 1–4). A new upper bound of 36 vertices for the least order of such cubic graphs is established. This bound is realized by cubic graphs without cut-vertices. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 177–182, 2001
TL;DR: A new class of algorithms is developed on the basis of a transformation of the graph problem to a geometric problem of clustering a set of points in Euclidean space to minimize the sum of the weights on edges joining Vertices in different parts of an edge-weighted undirected graph.
Abstract: This paper addresses the problem of partitioning the vertex set of an edge-weighted undirected graph into two parts of specified sizes so as to minimize the sum of the weights on edges joining Vertices in different parts. This problem is NP-hard and has several important applications in which the graph size is typically large and the brute-force approach (of listing all feasible partitions and comparing costs) is computationally prohibitive. In this paper a new class of algorithms is developed on the basis of a transformation of the graph problem to a geometric problem of clustering a set of points in Euclidean space. Instead of searching through all feasible partitions that meet the size specifications, it is shown that the search can be confined to a set of $n^{p( p + 1 )/2} $ partitions, where n is the number of vertices in the graph and p is the rank of the $n \times n$ graph connection matrix. Procedures are developed for constructing all such partitions in $O( n^{p( p + 3 ) /2} )$ time. For matrices...
07 Dec 1993
TL;DR: This shows that the method is able to divide a power system network model into a number of equal sized subnetworks in order to optimise the use of parallel computer systems for network analysis, while the resultant number of fill-in elements is kept to a minimum.
Abstract: An efficient heuristic algorithm for automatic partitioning of a power system network is presented. The algorithm exploits the techniques of factorization path graph partitioning and equivalent post-ordering. Test results for systems up to 811 busbars and 1476 transmission lines are included for comparison purposes. This shows that the method is able to divide a power system network model into a number of equal sized subnetworks in order to optimise the use of parallel computer systems for network analysis, while the resultant number of fill-in elements is kept to a minimum.
TL;DR: A point disconnecting set S of a graph G is a nontrivial m-separator, where m = |S|, if the connected components of G - S can be partitioned into two subgraphs, each of which has at least two points.
Abstract: A point disconnecting set S of a graph G is a nontrivial m-separator, where m = |S|, if the connected components of G - S can be partitioned into two subgraphs, each of which has at least two points. A 3-connected graph is quasi 4-connected if it has no nontrivial 3-separators. Suppose G is a graph having n ≥ 6 points. We prove three results: (1) If G is quasi 4-connected with at least 3n - 4 edges, then the graph K−1, obtained from K6 by deleting an edge, is a minor of G. (2) If G has at least 3n - 4 edges then either K−6 or the graph obtained by pasting two disjoint copies of K5 together along a triangle is a minor of G. (3) If the minimum degree of G is at least 6, then K−6 is a minor of G. © 1993 John Wiley & Sons, Inc.
TL;DR: The neighbourhoodsubgraph of the 7- vertices, which is the subgraph induced by the neighbour vertices of a vertex of degree 7, is investigated; they present 124 different types.
02 Jan 1993
TL;DR: In this article, it was shown that the group path problem is NP-complete even for bipartite graphs and cyclic groups, and presented a linear algorithm for this problem on chordal graphs.
Abstract: An integer sequence is a (graphical) degree sequence if and only if it is majorized by some threshold sequence. Therefore every degree sequence d can be transformed into a threshold sequence by a finite number (possibly zero) of reverse unit transformations. The minimum number of such transformations required is defined as the majorization gap of d. We prove a simple formula for the majorization gap by establishing a lower bound for it and exhibiting reverse unit transformations that achieve the bound. In addition, we characterize the degree sequences with maximum majorization gap.
Assume that each edge of a graph G is given a weight, which is an element of some group. The weight of a path is the product of the weights of the edges along the path. Given two vertices s and t and a group element g, the group path problem is to test if G contains a chordless s-t path of weight g. We prove that the group path problem is NP-complete even for bipartite graphs and cyclic groups, and present a linear algorithm for this problem on chordal graphs.
When the group is the additive group mod 2 and each edge has a unit weight, the group path problem becomes the NP-complete parity path problem, studied in connection with the Strong Perfect Graph Conjecture. A graph is called perfectly orientable if its edges can be oriented so that no chordless path on four vertices has both of its extreme edges oriented towards the middle edge. The class of perfectly orientable graphs properly contains the class of perfectly orderable graphs, studied in perfect graph theory. We present a polynomial algorithm for the parity path problem on perfectly orientable graphs.
01 Jan 1993
TL;DR: In this article, it was shown that if G is connected but not complete, then λ n(G)≤λ n(K_(n-1)~1)
Abstract: Let G be a simple graph with n vertices and λ_n(G) be the least eigenvalue of G.
In this paper, we show that, if G is connected but not complete, then λ_n(G)≤λ_n(K_(n-1)~1)
and the equality holds if and only if G K_(n-1)~1, where K_(n-1)~1, is the graph obtained by the
coalescence of a complete graph K_(n-1) of n-1 vertices with a path P_2 of length one of its
vertices.
TL;DR: A special clustering problem is discussed in this paper, called the compact set problem, which is to find all compact sets in a complete, weighted, undirected graph withn vertices.
Abstract: A special clustering problem is discussed in this paper, called the compact set problem. The goal of the problem is to find all compact sets in a complete, weighted, undirected graph withn vertices. A subsetC of vertices is called a compact set if 1<|C|
TL;DR: In this article, it was shown that the minimal connected generalized knight's graphs are planar or toroidal, which was unexpected in view of the complex nature of the graph generated by knight's moves on a chess-board.
Abstract: This paper is concerned with connectivity of graphs associated with generalized knight's moves. The minimal connected generalized knight's graphs are shown to be planar or toroidal, which was unexpected in view of the complex nature of the graph generated by knight's moves on a chess-board. Each move of a knight in chess takes the knight two steps parallel to one side of the board and one step parallel to the other side. Using moves of this kind a knight can move between any two positions on a standard 8 by 8 chess-board. Indeed it can move between any two positions on a 3 by 4 board, but not on any smaller board. A metrical geometry on the digital plane determined by standard knight's moves has been studied recently [1,6], and further metrics associated with generalized knight's moves have been noted [2]. The first step is to determine which knight's moves are transitive on an unbounded board. It is then natural to try to determine for each of these the smallest board on which it is transitive. It is convenient to express the problem in terms of knight's graphs in which the vertices are points in the plane with integer coefficients and the edges correspond to the knight's moves. An edge in the graph of a {p, ^}-knight joins two vertices where the difference between one of their coordinates is p and the difference between the other is q. Transitivity of a knight's moves on a board corresponds to connectivity of a knight's graph. The problem on the unrestricted set of vertices is easily solved. It is shown in §2 that the {p, ^}-knight's graph on Z 2 is connected if and only if p and q are mutually prime and their sum is odd. This is a special case of more general results on knight's graphs on Z" which have been studied by G. A. Jones [3]. The problem on restricted sets of vertices is more difficult. In this paper it is shown that if the {p, q)-knight's graph is connected on Z 2 and p
TL;DR: It is shown that a finite graph G on m vertices is isomorphic to a unit-distance graph in Rn if and only if there exists a real number r and for arbitrarily large integer r the graph G can be drawn in the n-dimensional integer lattice Zn.