Showing papers on "Path graph published in 1990"
TL;DR: For x and y vertices of a connected graph G, let TG(x, y) denote the expected time before a random walk starting from x reaches y, and it is determined that for each n > 0, the n‐vertex graph G and verticesx and y for whichTG(x) is maximized.
Abstract: For x and y vertices of a connected graph G, let TG(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n‐vertex graph G and vertices x and y for which TG(x, y) is maximized. the extremal graph consists of a clique on ⌊(2n + 1)/3⌋) (or ⌈)(2n − 2)/3⌉) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time TG(x, y) to reach y from x in this graph is approximately 4n3/27. © 1990 Wiley Periodicals, Inc.
129 citations
TL;DR: It is shown that either modulo (≤3)-separations, G can be drawn in a disc with no crossings except in one “small” area, and with its special vertices on the outside in the correct order.
113 citations
TL;DR: This work presents a linear algorithm for this problem on interval graphs, given the adjacency lists of an interval graph with n vertices and m edges, that runs in O(m+n) time.
111 citations
TL;DR: It is easily seen that every rinite graph has such a partition, and hence by compact- ness so does any locally finite graph.
66 citations
TL;DR: It is shown that every connected, 3-γ-critical graph on more than 6 vertices has a Hamiltonian path.
Abstract: In this paper we show that every connected, 3-γ-critical graph on more than 6 vertices has a Hamiltonian path.
49 citations
TL;DR: This paper presents the results of the authors’ investigation of a combinatorial problem arising from the study of evolutionary trees, a problem of colouring vertices of a binary tree, by which the significance of the maximum parsimony principle for selecting evolutionary trees can be judged.
Abstract: This paper presents the results of the authors’ investigation of a combinatorial problem arising from the study of evolutionary trees. In graph theoretic terms it can be expressed as a problem of colouring vertices of a binary tree. For a given colouring of the pendant vertices of a binary tree there is a simple algorithm for assigning colours to internal vertices minimising the number of edges of the tree whose end vertices have differing colours. This minimal number is called the length of the tree. The question posed is: For given numbers of pendant vertices of assigned colours, how many trees of a particular length can be constructed on those vertices? This question is answered in two special cases. Answers to this problem are needed to establish the distribution of lengths of evolutionary trees, by which the significance of the maximum parsimony principle for selecting evolutionary trees can be judged.
46 citations
TL;DR: The behaviour of a certain game of reflecting numbers written at the vertices of a graph is investigated using properties of the action of a Weyl group, of a Kac-Moody algebra associated with the graph, on its apartment.
45 citations
TL;DR: The n-center of G is the subgraph induced by those vertices of G having minimum n-eccentricity and it is shown that every graph is the n- center of some graph.
Abstract: The Steiner distance of a set S of vertices in a connected graph G is the minimum size among all connected subgraphs of G containing S. For n ≥ 2, the n-eccentricity en(ν) of a vertex ν of a graph G is the maximum Steiner distance among all sets S of n vertices of G that contains ν. The n-diameter of G is the maximum n-eccentricity among the vertices of G while the n-radius of G is the minimum n-eccentricity. The n-center of G is the subgraph induced by those vertices of G having minimum n-eccentricity. It is shown that every graph is the n-center of some graph. Several results on the n-center of a tree are established. In particular, it is shown that the n-center of a tree is a tree and those trees that are n-centers of trees are characterized.
43 citations
TL;DR: It is proved that every simple 3-regular graph admits a PPDC consisting of paths of length three, which is equivalent to a perfect path double cover of a graph G on n vertices.
Abstract: A perfect path double cover (PPDC) of a graph G on n vertices is a family of n paths of G such that each edge of G belongs to exactly two members of and each vertex of G occurs exactly twice as an end of a path of . We propose and study the conjecture that every simple graph admits a PPDC. Among other things, we prove that every simple 3-regular graph admits a PPDC consisting of paths of length three.
37 citations
TL;DR: In this paper, it was shown that any ordered 7-tuple of knot types can be realized as the associated list of knots types of (c 1,…, c 7 ) with a tame embedding of K 4 in Euclidean 3-space.
34 citations
TL;DR: It is proved in this paper that every simple graph G admits a perfect path double cover (PPDC), i.e., a set of paths of G such that each edge of G belongs to exactly two of the paths.
Abstract: We prove in this paper that every simple graph G admits a perfect path double cover (PPDC), i.e., a set of paths of G such that each edge of G belongs to exactly two of the paths and each vertex of G is an end of exactly two of the paths, where a path of length zero is considered to have (identical) ends. This was conjectured by A. Bondy in 1988.
TL;DR: This paper presents two efficient algorithms for finding maximum cliques of an overlap graph when it is given in the form of a family of n intervals, where γ is the total sum of their sizes.
Abstract: Let F = {I1, I2,…,In} be a finite family of closed intervals on the real line. Two intervals Ij and Ik in F are said to overlap each other if they intersect but neither one of them contains the other. A graph G = (V, E) is called an overlap graph for F if there is a one-to-one correspondence between V and F such that two vertices in V are adjacent to each other if and only if the corresponding intervals in F overlap each other. In this paper, we present two efficient algorithms for finding maximum cliques of an overlap graph when it is given in the form of a family of n intervals. The first algorithm finds a maximum clique in O (n. log n + Min {m, n- ω}) time, where m is the number of edges and ω is the size of a maximum clique, respectively, of the graph. The second algorithm generates all maximum cliques in O (n - log n + m + γ) time, where γ is the total sum of their sizes.
01 Jan 1990
TL;DR: A data structure that supports insertion and regular path existence queries inO(nk2) amortized time andO(|F|) worst-case time, respectively is presented and it is shown how this data structure and the techniques used for building it are applicable to the area of knowledge base querying.
Abstract: If $G$ is a directed graph with labeled edges and $L$ is a fixed regular language, the {\em regular path problem}, given two nodes, $u$ and $v$, in $G$, is to find a path between $u$ and $v$ such that the labels on the arcs along that path form a string which is a member of $L$. We consider a dynamic version of this problem, adding arcs to and performing regular path queries on $G$ over $L$, and present a data structure that solves both problems in average time per operation linear in the number of nodes of the graph for any fixed regular language.
01 Aug 1990
TL;DR: It is shown that there are, up to a trivial equivalence, precisely six theorems of the following form: If the vertices of a graph G are coloured red and white in such a way that no chordless path with four vertices is coloured in certain ways, then G is perfect if and only if each of the two subgraphs of G induced by all the Vertices of the same colour is perfect.
TL;DR: In this article, the Strong Perfect Graph Conjecture holds for graphs with triangle-free partner graphs, and it is shown that the partner graph of G is completely determined by the P4-structure of G.
TL;DR: It is proved that for almost all simple graphs over n vertices one needs Ω(n 4/3(log n)−4/3) extra vertices to obtain them as a double competition graph of a digraph.
Abstract: Here it is proved that for almost all simple graphs over n vertices one needs Ω(n4/3(log n)−4/3) extra vertices to obtain them as a double competition graph of a digraph. on the other hand O(n5/3) extra vertices are always sufficient. Several problems remain open.
TL;DR: A graph-theory model of synthons is suggested, in which the constraint of strict stoichiometry is removed and the virtual vertices formally correspond to functional groups that are not closely specified.
Abstract: A graph-theory model of synthons is suggested. A synthon is a special kind of the molecular graph in which some vertices are distinguished from other ones, and they are called the virtual vertices. The most important property of the synthons is that the constraint of strict stoichiometry is removed and the virtual vertices formally correspond to functional groups that are not closely specified.
TL;DR: In this paper, the graphical properties of the eigenvectors of adjacency matrix of a chemical graph are analyzed using the net-sign approach, and the topological contents stored in these eigen-vectors are described using vertex signed graphs (VSGs) and edge-signed graphs (ESGs).
Abstract: The graphical properties of the eigenvectors of adjacency matrix of a chemical graph are analysed using the net-sign approach. Model systems studied are G 65 , and branched-G 65 , where the subscripts represent the number of vertices and the number of edges. G 65 is the path with six vertices, hence five edges, isomorphic with the hydrogen-depleted graph of n-hexane, and branched-G 65 corresponds analogously to 3-methylpentane. The topological contents stored in these eigen-vectors are described using vertex-signed graphs (VSGs) and edge-signed graphs (ESGs). The relative ordering of net signs of ESGs is found to be similar to that of eigenvalues of the adjacency matrix. This simple analysis is also applied to polyenes and vinyl compounds.
TL;DR: It is shown that if m ⩽ 2 is an even integer and G is a graph such that dG(v) ⩾ m + 1 for all vertices v in G, then the line graph L(G) of G has a 2m-factor.
TL;DR: It is proved that ifG is a graph onn vertices of diameter 3 with maximum degreeD, D > 2.31 and it has the mimimum number of edges, then it is a porcupine.
Abstract: The graphG is called a porcupine, ifG∣A is a complete graph for some setA, every other vertex has degree one, and its only edge is joined toA. In this paper a conjecture of Bollobas is settled almost completely. Namely, it is proved that ifG is a graph onn vertices of diameter 3 with maximum degreeD, D > 2.31
$$\sqrt n $$
,D ≠ (n − 1)/2 and it has the mimimum number of edges, then it is a porcupine.
TL;DR: The implementation is not likely to be improved significantly without the improvement of the shortest paths algorithm and the minimum spanning tree algorithm as the algorithm essentially composes of the computation of the multiple sources shortest paths of a graph with ¦Vg−S¦ vertices and ¦Eg¦ edges.
Abstract: This paper studies the design and implementation of an approximation algorithm for the Steiner tree problem. Given any undirected distance graph G and a set of Steiner points S, the algorithm produces a Steiner tree with total weight on its edges no more than 2(1−1/L) times the total weight on the optimal Steiner tree, where L is the number of leaves in the optimal Steiner tree. Our implementation of the algorithm, in the worst case, makes it run in 0(¦Eg¦+¦Vg−S¦log¦Vg−S¦+¦S¦log ¦S¦) time for general graph G and in 0(¦S¦ log¦S¦+M log β(M,¦Vg−S¦)) time for sparse graph G, where Eg is the set of edges in G, Vg is the set of vertices in G, M = min {¦Eg, (¦Vg−S¦−1)2/2} and β(x,y) = min {i¦log(i)y ≦ x/y}.The implementation is not likely to be improved significantly without the improvement of the shortest paths algorithm and the minimum spanning tree algorithm as the algorithm essentially composes of the computation of the multiple sources shortest paths of a graph with ¦Vg¦ vertices and ¦Eg¦ edges and the minimum spanning tree of a graph with ¦Vg−S¦ vertices and M edges.
TL;DR: In this article, the traceability of polyhexes is examined and some practical guidelines for finding those that are not traceable are developed for finding simpler spannng subgraphs of the polyhex that will often make its traceability more obvious.
Abstract: The traceability of some of the smaller polyhexes is examined. (A graph is said to be traceable or to have a Hamiltonian path if it has a path visiting every vertex just once.) Most polyhexes are traceable, and an attempt is made to develop some practical guidelines for finding those that are not. A subgraph consisting of the branching vertices of a polyhex, and of any edges which join pairs of such vertices, is a useful tool for this purpose. The “principal resonance structures” of such a graph suggest ways of finding simpler spannng subgraphs of the polyhex that will often make its traceability, or lack of it, more obvious.
TL;DR: It is proved that, when k is fixed, a k-cut of almost every graph from $\mathcal{G}_{n,p} $ consists of $k - 1$ isolated vertices and one component on the remaining $n - k + 1$ vertices.
Abstract: The k-cut problem is to find a partition of a graph into k nonempty components, such that the number of edges between components is minimum. A random graph in $\mathcal{G}_{n,p}$ is a simple graph on n vertices with each pair of vertices connected by an edge with probability p. It is proved that, when k is fixed, a k-cut of almost every graph from $\mathcal{G}_{n,p} $ consists of $k - 1$ isolated vertices and one component on the remaining $n - k + 1$ vertices. An important outcome of this property is a linear algorithm that derives the minimum k-cut in such graphs, almost certainly.
TL;DR: The maximum number of cutvertices in a connected graph of order n having minimum degree at least δ is determined for δ ⩾5.
TL;DR: In this article, the authors present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form.
Abstract: From proposed mechanisms for framework reorganizations of the carboranes C2B
n-2H
n
,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.
TL;DR: It is proved that every graph has a code of length O(log|V|), which is distinct for every vertex υ ∈ V.
Abstract: Let G = (V, E) be an undirected graph with vertex set V and edge set E, and suppose that G does not contain isolated vertices and isolated edges. An assignment f : E → {0, 1} m is called a code of length m if the Boolean sum (or, alternatively, the mod 2 sum)
$$ f(\upsilon ): = \sum\limits_{e \in E,\upsilon \in e} {f(e)} $$
is distinct for every vertex υ ∈ V. We prove that every graph has a code of length O(log|V|).
TL;DR: It is proved that if m is odd then a partial m - cycle system on n vertices can be embedded in an m -cycle system on at most m (( m − 2) n ( n − 1) + 2 n + 1) vertices.
01 Nov 1990
TL;DR: It is proved that the all pairs shortest path problem can be solved by the repeated “plus-min” algorithm for computing the closure of the distance matrix of the graph in O((n3/p)log log n) average time by p, 1 ≤ p ≤n3.
Abstract: We consider the all pairs shortest path problem in a directed complete graph with n vertices, whose edge distances are non-negative random variables on parallel computers. We prove that the all pairs shortest path problem can be solved by the repeated “plus-min” algorithm for computing the closure of the distance matrix of the graph in O((n3/p)log log n) average time by p, 1 ≤ p ≤n3, processors on an SIMD-SM-RW computer for the graph whose edge distances are independent random variables drawn from [0, +∞] according to an arbitrary identical probability distribution.
01 Jan 1990
TL;DR: In this paper, it was shown that the circuits are the only connected graphs for which the equality n(G,k) = m(G/k) is satisfied for all values of k.
Abstract: Let the numbers of k-element sets of independent vertices and edges of a graph G be denoted by n(G,k) and m(G,k), respectively. Some properties of the numbers n(G,k) and m(G,k) are outlined. In particular, it is shown that the circuits are the only connected graphs for which the equality n(G,k) = m(G,k) is satisfied for all values of k.