Showing papers on "Path graph published in 1991"
TL;DR: An algorithm is presented that computes the visibility graph of s set of obstacles in time O(E + n log n), where E is the number of edges in the visibilitygraph and n is the total number of vertices in all the obstacles.
Abstract: The visibility graph of a set of nonintersecting polygonal obstacles in the plane is an undirected graph whose vertex set consists of the vertices of the obstacles and whose edges are pairs of vertices $(u,v)$ such that the open line segment between u and v does not intersect any of the obstacles. The visibility graph is an important combinatorial structure in computational geometry and is used in applications such as solving visibility problems and computing shortest paths. This paper presents an algorithm that computes the visibility graph of a set of obstacles in time $O(E + n\log n)$, where E is the number of edges in the visibility graph and n is the total number of vertices in all the obstacles.
243 citations
TL;DR: It is shown that for both variants of the game, the problem of determining whether there is a winning strategy for player 1 is PSPACE-complete for any C with |C|≥3, but the problems are solvable in , and time, respectively, if |C |=2.
Abstract: In this paper we consider the following game: players must alternately color the lowest numbered uncolored vertex of a given graph G= ({1, 2,…, n}, E) with a color, taken from a given set C, such that two adjacent vertices are never colored with the same color. In one variant, the first player which is unable to move, loses the game. In another variant, player 1 wins the game if and only if the game ends with all vertices colored. We show that for both variants, the problem of determining whether there is a winning strategy for player 1 is PSPACE-complete for any C with |C|≥3, but the problems are solvable in , and time, respectively, if |C|=2. We also give polynomial time algorithms for the problems with certain restrictions on the graphs and orderings of the vertices. We give some partial results for the versions where no order for coloring the vertices is specified.
213 citations
TL;DR: The following problems, of possible interest with regards to perfect graphs, are shown to be NP-Complete.
151 citations
01 Sep 1991
TL;DR: Ambivalent data structures are presented for several problems on undirected graphs and used in finding the k smallest spanning trees of a weighted undirecting graph in O(m log beta (m,n)+min(k/sup 3/2/, km/sup 1/2/)) time, where m and n are understood to be the current number of edges and vertices, respectively.
Abstract: Ambivalent data structures are presented for several problems on undirected graphs. They are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log beta (m,n)+min(k/sup 3/2/, km/sup 1/2/)) time, where m is the number of edges and n the number of vertices in the graph. The techniques are extended to find the k smallest spanning trees in an embedded planar graph in O(n+k(log n)/sup 3/) time. Ambivalent data structures are also used to maintain dynamically 2-edge-connectivity information. Edges and vertices can be inserted or deleted in O(m/sup 1/2/) time, and a query as to whether two vertices are in the same 2-edge-connected component can be answered in O(log n) time, where m and n are understood to be the current number of edges and vertices, respectively. Again, the techniques are extended to maintain an embedded planar graph so that edges and vertices can be inserted or deleted in O((log n)/sup 3/) time, and a query answered in O(log n) time. >
112 citations
TL;DR: In this article, a linear-time algorithm is presented for finding an appropriate embedding of a directed planar graph G and a corresponding face-on-vertex covering of cardinality O(p), where p is the minimum cardinality of a subset of the faces that cover all vertices.
Abstract: An algorithm is presented for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with real-valued edge costs but no negative cycles. The algorithm runs in O(pn) time, where n is the number of vertices in G, and p is the minimum cardinality of a subset of the faces that cover all vertices, taken over all planar embeddings of G. The algorithm is based on a decomposition of the graph into O(pn) outerplanar subgraphs satisfying certain separator properties. Linear-time algorithms are presented for various subproblems including that of finding an appropriate embedding of G and a corresponding face-on-vertex covering of cardinality O(p), and of generating all pairs shortest path information in a directed outerplannar graph.
82 citations
01 Sep 1991
TL;DR: In this paper, the problem of performing k-connectivity queries for k > is considered and the authors consider a graph G with n vertices and m edges, and ask whether there exist k disjoint paths between vertices v' and v' of G.
Abstract: Given a graph G with n vertices and m edges, a k-connectivity query for vertices v' and v" of G asks whether there exist k disjoint paths between v' and v". The authors consider the problem of performing k-connectivity queries for k >
58 citations
01 Sep 1991
TL;DR: For traversing layered graphs consisting of w disjoint paths tied together at a common source, the authors give a randomized online algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor.
Abstract: A layered graph is a connected, weighted graph whose vertices are partitioned into sets L/sub 0/=(s), L/sub 1/, L/sub 2/, . . ., and whose edges run between consecutive layers. Its width is max( mod L/sub i/ mod ). In the online layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i-1 and i are only revealed when the searcher reaches layer i-1. The authors give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. They give a deterministic online algorithm that is O(9w)-competitive on width-w graphs and prove that for no w can a deterministic online algorithm have a competitive ratio better than 2w/sup -2/ on width-w graphs. They prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized online layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, they give a randomized online algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor. >
57 citations
Proceedings Article•
01 Jan 1991
TL;DR: The authors present a static data structure that answers k-connectivity queries for k
Abstract: Given a graph G with n vertices and m edges, a kconnectivity query for vertices v' and U'' of G asks whether there exist k disjoint paths between U' and U". Answering such queries has important applications to network reliability. In this paper we consider the problem of performing k-connectivity queries for k 5 4. First, we present a static data structure that answers such queries in 0(1) time. Next, we consider the problem of performing queries intermixed with on-line updates that insert vertices and edges. For triconnected graphs we give a dynamic data structure that supports queries and updates in time O(a(e,n)) amortized, where n is the current number of vertices of the graph and e is the total number of operations performed (a(!, n) denotes the slowly growing Ackermann's function inverse). For general graphs, a sequence of e operations takes total time O(n1ogn + e). AI1 of the above data structures use space O(n), proportional to the number of vertices of the graph. Our results also yield an eficient algorithm for testing whether graph G is four-connected that runs in O(n a(n, n) + m) time using O(n + m) space.
52 citations
TL;DR: It is shown that in the class of allKr-free graphs withn vertices the complete balanced (r − 1)-partite graphTr−1(n) has the largest number of subgraphs isomorphic toKt (t < r),C4,K2,3.
Abstract: Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) ≤ H(T 2(n)), whereT 2(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n ? max(m, r ? 1), then there exists a complete (r ? 1)-partite graphG* withn vertices such thatK(G) ≤ K(G*) holds for everyK r -free graphG withn vertices. In particular, in the class of allK r -free graphs withn vertices the complete balanced (r ? 1)-partite graphT r?1(n) has the largest number of subgraphs isomorphic toK t (t < r),C 4,K 2,3. These generalize some theorems of Turan, Erdos and Sauer.
52 citations
TL;DR: Let G be a k-connected graph with minimum degree d and at least 2d vertices and G has a cycle of length at least 1d through any specified set of k vertices.
36 citations
TL;DR: In this article, it was shown that any graph of n vertices and tr−1(n)+m edges, where tr− 1(n) is the Turan number, contains (1−o(1)m edge disjoint kr's ifm = o(n 2 ).
Abstract: In this paper, we prove that any graph ofn vertices andtr−1(n)+m edges, wheretr−1(n) is the Turan number, contains (1−o(1)m edge disjointKr'sifm=o(n2). Furthermore, we determine the maximumm such that every graph ofn vertices andtr−1(n)+m edges containsm edge disjointKr's ifn is sufficiently large.
TL;DR: An optimal O(n^2) algorithm is obtained that finds the degree of connectedness between all pairs of distinct vertices in the graph.
TL;DR: A polynomial characterization of label-connected graphs with n vertices and 2n − 4 edges is obtained and the following problem is established: Given a graph G and two vertices u and v of G, does there exist a (u, v)-path P in G such that G−E(P) is connected?
TL;DR: It is shown that there are only six other expansions of trees that are symmetric, four of which are Y -graphs, expansions of K 1.3 , and two are H - graphs, expansion of the tree with two nodes ofdegree 3 and four nodes of degree 1.
TL;DR: It is proved that if t is a fixed positive integer and n is sufficiently large, then each graph of order n with minimum degree n − t has an assignment of weights 1, 2 or 3 to the edges in such a way that weighted degrees of the vertices become distinct.
TL;DR: This paper determines Г k (P n ) , the largest cardinality of a k -minimal dominating set of the n -vertex path P n and proves for any n - Vertex graph G, Г 2 (G)γ(G) ≤ n and a 'Gallai-type' theorem for k-minimal parameters is established.
TL;DR: A necessary and sufficient condition is obtained for a set of 12 vertices in any 3-connected cubic graph to lie on a common cycle.
Abstract: A necessary and sufficient condition is obtained for a set of 12 vertices in any 3-connected cubic graph to lie on a common cycle.
TL;DR: Let D be a strong digraph with n vertices and at least ( n − 1)( n − 2) + 3 arcs such that n = n 1 + n 2 +⋯+ n k and n i ⩾3, there is covering of the vertices of D by disjoint directed cycles of length n 1, n 2,…, n k.
TL;DR: A graph with n vertices and minimum degree k ⩾2 can contain no more than (2k−2)n (k 2 −2) cut vertices, and this bound is asymptotically tight.
TL;DR: G ishamiltonian or {x, y, z} is contained on a cycle of length at least 2d in G, which is a 2-connected graph with minimum degree d.
Abstract: Let Gbe a 2-connected graph with minimum degree d and let {x, y, z} be a set of three vertices contained on some cycle ofG. ThenG ishamiltonian or {x, y, z} is contained on a cycle of length at least 2d inG.
TL;DR: The minimum number of edges a graph needs in order to ensure that the subgraph induced by all vertices of degree at least d contains a cycle of length at least c or cycles of all lengths up to c is determined.
TL;DR: It is shown that an undirected graph with no infinite independent set is covered by finitely many pairwise disjoint paths, and it is proved that if to each edge (x, y) of a countable path is associated an element of a finite group, then some edges can be deleted so that the new graph is still a path.
15 Dec 1991-JSME international journal. Series 3, Vibration, control engineering, engineering for industry
TL;DR: In this article, a computer algorithm for determining the symmetry of a kinematic chain is developed, where the vertex set of the graph generated from the Kinematic Chain is partitioned into classes of vertices according to certain structural properties in graph theory.
Abstract: A computer algorithm for determining the symmetry of a kinematic chain is developed. The vertex set of the graph generated from the kinematic chain is partitioned into classes of vertices according to certain structural properties in graph theory. Permutations of vertices in each class are selected if they can map the set of edges into the original set of edges corresponding to the class. They are then concatenated with those from other classes. All the permutations that remain after checking the automorphism become the vertex-induced group of the graph, which represents the symmetry of the corresponding kinematic chain.
TL;DR: If r = log(nωn) where ωn → ∞, then almost all of these graphs have no isolated black vertices, are connected and have a perfect matching, so that a random digraph that is regular of outdegree r almost surely has a cycle cover.
TL;DR: In this paper, it was shown that for any two topologically similar vertices i and k, A2ji=A2jk; for all j≠k and i where Aji is the cofactor of the (j,i) element of the secular determinant det(xI-A), A is the adjacency matrix of the graph G and I is the (n×n) unit matrix.
Abstract: It has been proved that if {vi}, i= 1–n, is the set of vertices of an undirected labelled graph G, then for any two topologically similar vertices i and k, A2ji=A2jk; for all j≠k and i where Aji is the cofactor of the (j,i) element of the secular determinant det(xI–A), A is the adjacency matrix of the graph G and I is the (n×n) unit matrix.As any real symmetric matrix, A, can be represented by an undirected vertex- and edge-weighted graph (G), the above relation has been utilised, in conjunction with a recently developed graph-theoretical method for expressing eigenvectors of A as polynomials in terms of eigenvalues, to determine a good number of eigenvalues of the matrix. The method, for the first time, utilises a newly developed technique of determination of eigenvectors for evaluation of eigenvalues. In one particular case it has been shown that the present method can reduce the required polynomial equations to a degree lower than that possible by McClelland's technique for factorisation of chemical graphs. Some applications of the method (other than HMO theory), for example, calculation of principal stress tensors in fluid dynamics and force constants in a molecular vibration problem, are illustrated.
01 Jan 1991
TL;DR: Gobel and Veldman as mentioned in this paper showed that label-connected graphs are NP-complete and showed that for a graph G and two vertices u and v of G, there exists a (u, v)-path P in G such that G - E(P) is connected.
Abstract: Gobel, F., J. Orestes Cerdeira and H.J. Veldman, Label-connected graphs and the gossip problem, Discrete Mathematics 87 (1991) 29-40. A graph with m edges is called label-connected if the edges can be labeled with real numbers in such a way that, for every pair (u, v) of vertices, there is a (u, v)-path with ascending labels. The minimum number of edges of a label-connected graph on n vertices equals the minimum number of calls in the gossip problem for n persons, which is known to be 2n - 4 for n z 4. A polynomial characterization of label-connected graphs with n vertices and 2n - 4 edges is obtained. For a graph G, let 4(G) denote the minimum number of edges that have to be added to E(G) in order to create a graph with two edge-disjoint spanning trees. It is shown that for a graph G to be label-connected, $(G) c 2 is necessary and @(G) & 1 is sufficient. For i = 1, 2, the condition #(G) c i can be checked in polynomial time. Yet recognizing label-connected graphs is an NP-complete problem. This is established by first showing that the following problem is NP-complete: Given a graph G and two vertices u and v of G, does there exist a (u, v)-path P in G such that G - E(P) is connected?