Showing papers on "Path graph published in 1997"
TL;DR: Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of partitioning the vertices of a weighted graph intok blocks so as to maximize the weight of crossing edges.
Abstract: Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph intok blocks so as to maximize the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks of equal cardinality, again so as to maximize the weight of crossing edges. The approach, pioneered by Goemans and Williamson, is via a semidefinite programming relaxation.
400 citations
TL;DR: In this paper, the minimum number of colors required for vertex-distinguishing proper edge-coloring of a simple graph G is denoted by, where n denotes the number of vertices of degree i in G.
Abstract: An edge-coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge-coloring of a simple graph G is denoted by . A simple count shows that where ni denotes the number of vertices of degree i in G. We prove that where C is a constant depending only on Δ. Some results for special classes of graphs, notably trees, are also presented. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 73–82, 1997
165 citations
TL;DR: Ambivalent data structures are presented for several problems on undirected graphs and used to dynamically maintain 2-edge-connectivity information and are extended to find the smallest spanning trees in an embedded planar graph in time.
Abstract: Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the $k$ smallest spanning trees of a weighted undirected graph in $O(m \log \beta (m,n) + \min \{ k^{3/2}, km^{1/2} \} )$ 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 $O(n + k (\log n)^3 )$ time. Ambivalent data structures are also used to dynamically maintain 2-edge-connectivity information. Edges and vertices can be inserted or deleted in $O(m^{1/2})$ 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.
132 citations
05 Jan 1997
TL;DR: In this paper, the problem of finding small-stretch paths between all pairs of vertices in a weighted undirected graph with n vertices and m edges was studied.
Abstract: Let G = (V, E) be a weighted undirected graph. A path between u, v {element_of} V is said to be of stretch t if its length is at most t times the distance between u and v in the graph. We consider the problem of finding small-stretch paths between all pairs of vertices in the graph G. It is easy to see that finding paths of stretch less than 2 between all pairs of vertices in an undirected graph with n vertices is at least as hard as the Boolean multiplication of two n x n matrices. We describe three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges. The first algorithm, STRETCH{sub 2}, runs in O(n{sup 3}/{sup 2}m{sup {1/2}}) time and finds stretch 2 paths. The second algorithm, STRETCH{sub 7/3}, runs in O(n{sup 7/3}) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH{sub 3}, runs in O(n{sup 2}) and finds stretch 3 paths. Our algorithms are simpler, more efficient and more accurate than the previously best algorithms for finding small-stretch paths. Unlike all previous algorithms, our algorithms are not based on the construction of sparsemore » spanners or sparse neighborhood covers.« less
82 citations
TL;DR: It is shown that, for any fixed k ≥ 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graphs on n n verticeswith no k -1 pairwise crossing edges containing at mostO(n log n) edges.
Abstract: A {\em geometric graph\/} is a graph $G=(V,E)$ drawn in the plane so that the vertex set $V$ consists of points in general position and the edge set $E$ consists of straight line segments between points of $V$. Two edges of a geometric graph are said to be {\em parallel\/}, if they are opposite sides of a convex quadrilateral. In this paper we show that, for any fixed $k\ge3$, any geometric graph on $n$ vertices with no $k$ pairwise parallel edges contains at most $O(n)$ edges, and any geometric graph on $n$ vertices with no $k$ pairwise crossing edges contains at most $O(n\log n)$ edges. We also prove a conjecture of Kupitz that any geometric graph on $n$ vertices with no pair of parallel edges contains at most $2n-2$ edges.
73 citations
TL;DR: It is shown that if n>n0(d) then any d-regular graph G=(V, E) on n vertices contains a set of u=⌊n/2⌋ vertices which is joined by at most (d/2−c√d)u edges to the rest of the graph, where c>0 is some absolute constant.
Abstract: It is shown that if n>n0(d) then any d-regular graph G=(V, E) on n vertices contains a set of u=⌊n/2⌋ vertices which is joined by at most (d/2−c√d)u edges to the rest of the graph, where c>0 is some absolute constant. This is tight, up to the value of c.
67 citations
18 Sep 1997
TL;DR: It is shown that for any fixed r-colorable graph of n vertices, every three-dimensional grid drawing that fits into a box of volume O(n^2)$.
Abstract: A three-dimensional {\em grid drawing} of a graph $G$ is a placement of the vertices at distinct integer points so that the straight-line segments representing the edges of $G$ are pairwise non-crossing. It is shown that for any fixed $r\geq 2$, every $r$-colorable graph of $n$ vertices has a three-dimensional grid drawing that fits into a box of volume $O(n^2)$. The order of magnitude of this bound cannot be improved.
65 citations
TL;DR: It is proved that every 3-connected planar graph G either contains a path on k vertices each of which has degree at most 5k or does not contain any path onk vertices; the bound 5k is the best possible.
Abstract: We have proved that every 3-connected planar graph G either contains a path on k vertices each of which has degree at most 5k or does not contain any path on k vertices; the bound 5k is the best possible. Moreover, for every connected planar graph H other than a path and for every integer m ? 3 there is a 3-connected planar graph G such that each copy of H in G contains a vertex of degree at least m.
48 citations
TL;DR: This paper formulate efficient routines for greedy exchanges of paths as well as vertices for Steiner's problem in a weighted graph using a heuristic based on an initialization providing shortest-path information, a module for the selection of Steiner vertices, and an improvement procedure.
Abstract: Steiner's problem in a weighted graph requires a tree of minimum total weight spanning a subset of special vertices. In this paper, we formulate efficient routines for greedy exchanges of paths as well as vertices. The heuristic proposed on the basis of these routines consists of three phases: an initialization providing shortest-path information, a module for the selection of Steiner vertices, and an improvement procedure. The latter procedure constitutes a general procedure executable after any heuristic. A spin-off from the second module is a decreased running time for a well-known 11/6 approximation algorithm. The first phase can be replaced by a shortest-path approximation method to obtain a running time order that is quadratic in the number of vertices.
45 citations
TL;DR: If a 4-connected plane graph G has at least four vertices on its external face, then G can be embedded on a grid of size W × H such that W + H ≤n, W ≤ (n + 3)/2 and H ≤ 2(n - 1)/3, where n is the number of vertices of G.
Abstract: A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper we show that if a 4-connected plane graph G has at least four vertices on its external face, then G can be embedded on a grid of size W × H such that W + H ≤n, W ≤ (n + 3)/2 and H ≤ 2(n - 1)/3, where n is the number of vertices of G. Such an embedding can be computed in linear time.
44 citations
TL;DR: A simple linear-time algorithm to find a partition V1, V2, V3, V4 of the vertex set V such that ui ∈ Vi, ¦Vi¦=ni and Vi induces a connected subgraph of G for each i, 1 ≤ i ≤ 4 is given.
TL;DR: In this paper, it was shown that the vertices and edges of the Buneman graph can be described in a very simple way: given a collection of splitsS, S correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′.
Abstract: In 1971, Peter Buneman proposed a way to construct a tree from a collection of pairwise compatible splits. This construction immediately generalizes to arbitrary collections of splits, and yields a connected median graph, called the Buneman graph. In this paper, we prove that the vertices and the edges of this graph can be described in a very simple way: given a collection of splitsS, the vertices of the Buneman graph correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′. Using this characterization, it is much more straightforward to construct the vertices of the Buneman graph than using prior constructions. We also recover as an immediate consequence of this enumeration that the Buneman graph is a tree, that is, that the number of vertices exceeds the number of edges (by one), if and only if any two distinct splits inS are compatible.
TL;DR: A randomized (Las Vegas) algorithm is given for finding the Gallai--Edmonds decomposition of a graph that is significantly better than the best previous ones, e.g., using the best value of M(n) currently known, the new sequential running time is O(n2.5 /(log n) or more.
Abstract: A randomized (Las Vegas) algorithm is given for finding the Gallai--Edmonds decomposition of a graph. Let n denote the number of vertices, and let M(n) denote the number of arithmetic operations for multiplying two n $\times$ n matrices. The sequential running time (i.e., number of bit operations) is within a poly-logarithmic factor of M(n). The parallel complexity is O((log n)2) parallel time using a number of processors within a poly-logarithmic factor of M(n). The same complexity bounds suffice for solving several other problems:
finding a minimum vertex cover in a bipartite graph
finding a minimum X ---> Y vertex separator in a directed graph, where X and Y are specified sets of vertices,
finding the allowed edges (i.e., edges that occur in some maximum matching) of a graph,
finding the canonical partition of the vertex set of an elementary graph.
The sequential algorithms for problems (i), (ii), and (iv) are Las Vegas, and the algorithm for problem (iii) is Monte Carlo. The new complexity bounds are significantly better than the best previous ones, e.g., using the best value of M(n) currently known, the new sequential running time is O(n2.38) versus the previous best O(n2.5 /(log n)) or more.
TL;DR: In this article, it was shown that for almost all connected graphs with minimum degree at least 2 and q edges, the domination number is bounded by (q + 1)/3.
Abstract: A dominating set for a graph G = (V,E) is a subset of vertices V′ ⊆ V such that for all v E V − V′ there exists some u E V′ for which {v, u} E E. The domination number of G is the size of its smallest dominating set(s). We show that for almost all connected graphs with minimum degree at least 2 and q edges, the domination number is bounded by (q + 1)/3. From this we derive exact lower bounds for the number of edges of a connected graph with minimum degree at least 2 and a given domination number. We also generalize the bound to k-restricted domination numbers; these measure how many vertices are necessary to dominate a graph if an arbitrary set of k vertices must be incluced in the dominating set. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 139–152, 1997
TL;DR: In this article, it was shown that every k-hypertournament on n (k) vertices has a Hamiltonian path (an extension of Redeis theorem on tournaments).
Abstract: Given two integers n and k, n ≥ k > 1, a k-hypertournament T on n vertices is a pair (V, A), where V is a set of vertices, |V| = n and A is a set of k-tuples of vertices, called arcs, so that for any k-subset S of V, A$ contains exactly one of the k! k-tuples whose entries belong to S. A 2-hypertournament is merely an (ordinary) tournament. A path is a sequence v1a1v2v3···vt−1vt of distinct vertices v1, v2,⋖, vt and distinct arcs a1, ⋖, at−1 such that vi precedes vt−1 in a, 1 ≤ i ≤ t − 1. A cycle can be defined analogously. A path or cycle containing all vertices of T (as vi's) is Hamiltonian. T is strong if T has a path from x to y for every choice of distinct x, y ≡ V. We prove that every k-hypertournament on n (k) vertices has a Hamiltonian path (an extension of Redeis theorem on tournaments) and every strong k-hypertournament with n (k + 1) vertices has a Hamiltonian cycle (an extension of Camions theorem on tournaments). Despite the last result, it is shown that the Hamiltonian cycle problem remains polynomial time solvable only for k ≤ 3 and becomes NP-complete for every fixed integer k ≥ 4. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 277–286, 1997
17 Dec 1997
TL;DR: It is shown that a graph is planar if and only if it is isomorphic to a diamond-free pseudo-line graph and the O(k⅓n) upper bound on the k-level complexity of an arrangement of straight-lines, which is very recently discovered by Dey, holds for an arrangements of pseudo-lines as well.
Abstract: Let Γ be an arrangement of pseudo-lines, i.e., a collection of unbounded x-monotone curves in which each curve crosses each of the others exactly once. A pseudo-line graph (ΓT, E) is a graph for which the vertices are the pseudo-lines of Γ and the edges are some un-ordered pairs of pseudo-lines of Γ. A diamond of pseudo-line graph (Γ, E) is a pair of edges p, q), p′, q′) ∈ E, (p′, q′) ∩ p′, q′ = 0, such that the crossing point of the pseudo-lines p and q lies vertically between p′ and q′ and the crossing point of p′ and q′ lies vertically between p and q. We show that a graph is planar if and only if it is isomorphic to a diamond-free pseudo-line graph. An immediate consequence of this theorem is that the O(k⅓n) upper bound on the k-level complexity of an arrangement of straight-lines, which is very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.
TL;DR: It is shown that the Wiener number of an n-hexagonal net equals 1 5 (164n 5 − 30n 3 + n) .
27 Feb 1997
TL;DR: A concurrent-read exclusive-write PRAM algorithm is developed to find the k shortest paths between pairs of vertices in an edge-weighted directed graph, where L is the total length of the output.
Abstract: A concurrent-read exclusive-write PRAM algorithm is developed to find the k shortest paths between pairs of vertices in an edge-weighted directed graph. Repetitions of vertices along the paths are allowed. The algorithm computes an implicit representation of the k shortest paths to a given destination vertex from every vertex of a graph with n vertices and m edges, using O(m + nk log2k) work and O(log3k log*k+ log n(log log k+ log*n)) time, assuming that a shortest path tree rooted at the destination is precomputed. The paths themselves can be extracted from the implicit representation in O(log k+log n) time, and O(n log n + L) work, where L is the total length of the output.
18 Sep 1997
TL;DR: Two algorithms for orthogonal graph drawing in three dimensional space are presented that guarantee no crossings and can be used under an interactive setting (i.e., vertices arrive and enter the drawing on-line), as well.
Abstract: We present two algorithms for orthogonal graph drawing in three dimensional space For graphs of maximum degree six, the 3-D drawing is produced in linear time, has volume at most 466n3 and each edge has at most three bends If the degree of the graph is arbitrary, the vertices are represented by solid 3-D boxes whose surface is proportional to their degree The produced drawing has two bends per edge Both algorithms guarantee no crossings and can be used under an interactive setting (ie, vertices arrive and enter the drawing on-line), as well
TL;DR: The results prove that in a graph G × P, where P is a path with endvertices x and y, and G is any graph, with vertices a and b, the resistance between vertices (a, x) and (b, v) is maximised at v = y.
TL;DR: The method to obtain p-central vertices, which are vertices that can be used to re-routing any communication affected by the faulty elements, is given, which implies that the diameter of the corresponding surviving route graph is optimum.
TL;DR: It is shown that a 1-heavy or a 2-heavy graph G has a Hamilton cycle or a perfect matching if the authors impose certain additional conditions on G involving numbers of common neighbours, (local) connectivity, and forbidden induced subgraphs.
TL;DR: It is shown that [ n 3 ] edge guards are always sufficient to cover a triangulated polyhedral terrain on n vertices by showing that [ 2n 5 ] edges are necessary to cover all of the faces of an arbitrary plane graph on n Vertices.
Abstract: In this paper we show that [ n 3 ] edge guards are always sufficient to cover a triangulated polyhedral terrain on n vertices We prove this by showing that [ n 3 ] edges are always sufficient to cover all of the faces of a plane triangulation on n vertices We also show that [ 2n 5 ] edges are sufficient to cover all of the faces of an arbitrary plane graph on n vertices
TL;DR: This paper finds the greatest number of edges of a highly irregular graph with n vertices, where n is an odd integer (for n even this number is given in [1]), and the smallest number of edge of ahighly irregular graph of given order.
TL;DR: This is a new lower bound for the circumference c(G) of a 3-connected graph G, where ?3 = min{{?i=13 d(ui) ?
TL;DR: This work completely characterize the class of minimum self-repairing graphs, which have the fewest edges for a given number of vertices.
Abstract: A graph is self-repairing if it is 2-connected and such that the removal of any single vertex results in no increase in distance between any pair of remaining vertices of the graph. We completely characterize the class of minimum self-repairing graphs, which have the fewest edges for a given number of vertices.
TL;DR: This paper presents an O(mn3)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one.
Patent•
06 Oct 1997TL;DR: In this paper, a method for operating a computer to find elementary loops in a strongly connected component of a graph is presented, where the computer identifies a starting vertex from the vertices of the strongly connected components that have not been examined as a possible starting vertex for an elementary loop and which are not contained in any elementary loop discovered thus far.
Abstract: A method for operating a computer to find elementary loops in a strongly connected component of a graph. In the basic method, the computer identifies a starting vertex from the vertices of the strongly connected component that have not been examined as a possible starting vertex for an elementary loop and which are not contained in any elementary loop discovered thus far. The vertices of the strongly connected component are then searched for a path starting and ending on the identified vertex. If the search finds a path starting and ending of the identified vertex, the path is recorded as an elementary loop. This process is repeated until no starting vertex can be identified. To improve the efficiency of the search process, the computer identifies vertices that are the starting vertex for a paths that are shared by more than one elementary loop. The shared paths are stored separately and used to avoid searching the vertices of the path more than once.
TL;DR: A characterization of those values of n and m for which there exists a connected graph with n vertices and m edges that has no spanning star tree, and a proof that finding spanning star trees is in general NP-hard.