Showing papers on "Path graph published in 1994"
TL;DR: This note concerns the problem of finding the sparsest 2-spanner in a given graph and presents an approximation algorithm for this problem with approximation ratio log(|E|/|V|)
135 citations
TL;DR: The relationship between λ(G) and another graph invariant, the path covering number of Gc, is derived and applications include the determination of the λ-number of the join of two graphs, the product of two complete graphs, and the complete multi-partite graphs.
128 citations
TL;DR: An algorithm for vertex-coloring graphs is said to be online if each vertex is irrevocably assigned a color before any later vertices are considered, but it is shown that such algorithms are inherently ineffective.
88 citations
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TL;DR: It is proved that the maximal number of edges in a graph with n ≧ 8 vertices that is not contractible to K8 is 6n − 21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6 n − 21 edges that are not contractable to K9 are the K5(2)-cockades.
Abstract: It is proved that the maximal number of edges in a graph with n ≧ 8 vertices that is not contractible to K8 is 6n − 21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6n − 21 edges that are not contractible to K8 are the K5(2)-cockades that have exactly 6n − 20 edges.
83 citations
20 Nov 1994
TL;DR: The problem is a simple abstraction of a robot motion planning problem, with the geometry replaced by the adjacencies in the graph, and its complexity is studied, giving exact and approximate algorithms for several cases.
Abstract: We are given a connected, undirected graph G on n vertices. There is a mobile robot on one of the vertices; this vertex is labeled s. Each of several other vertices contains a single movable obstacle. The robot and the obstacles may only reside at vertices, although they may be moved across edges. A vertex may never contain more than one object (robot/obstacle). In one step, we may move either the robot or one of the obstacles from its current position /spl upsi/ to a vacant vertex adjacent to v. Our goal is to move the robot to a designated vertex t using the smallest number of steps possible. The problem is a simple abstraction of a robot motion planning problem, with the geometry replaced by the adjacencies in the graph. We point out its connections to robot motion planning. We study its complexity, giving exact and approximate algorithms for several cases. >
64 citations
TL;DR: It is shown that every geometric graph withn vertices andm>k4n edges containsk+1 pairwise disjoint edges, and it is proved that, given a set of pointsV and aSet of axis-parallel rectangles in the plane, then either there arek-1 rectangles such that no point ofV belongs to more than one of them, or the authors can find an at most 2·105k8 element subset ofV meeting all rectangles.
Abstract: A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear. We settle an old question of Avital, Hanani, Erd?s, Kupitz, and Perles by showing that every geometric graph withn vertices andm>k4n edges containsk+1 pairwise disjoint edges. We also prove that, given a set of pointsV and a set of axis-parallel rectangles in the plane, then either there arek+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most 2·105k8 element subset ofV meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth's theorem on partially ordered sets.
63 citations
10 Jun 1994
TL;DR: It is shown that any graph of n vertices that can be drawn in the plane with no k+1 pairwise crossing edges has at most c-c-k-n log 2-supscrpt-−2 edges, which gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdo&huml;s, Kupitz, Perles, and others.
Abstract: We show that any graph of n vertices that can be drawn in the plane with no k+1 pairwise crossing edges has at most cknlog2k−2n edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdős, Kupitz, Perles, and others. We also construct two point sets {p1,…,pn}, {q1,…,qn} in the plane such that any piecewise linear one-to-one mapping f:R2→R2 with f(pi)=qi (1≤i≤n) is composed of at least Ω(n2) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.
63 citations
TL;DR: A graph X is called 1/2-transitive if the automorphism group of X acts transitively on the set of vertices, edges, or arcs of X, respectively as mentioned in this paper.
Abstract: A graph X is called vertex-transitive, edge-transitive, or arc-transitive, if the automorphism group of X acts transitively on the set of vertices, edges, or arcs of X, respectively. X is said to be 1/2-transitive, if it is vertex-transitive, edge-transitive, but not arc-transitive.
In this paper we determine all 1/2-transitive graphs with 3p vertices, where p is an odd prime. (See Theorem 3.4.)
59 citations
TL;DR: It is proved that a graph on the torus is 5-colorable, unless it contains either K6 the complete graph on six vertices, or C3 + C5, the join of two cycles of lengths three and five, respectively.
53 citations
TL;DR: It is shown that there is a graph L with n vertices and at least n1.36 edges such that it contains neither L 3 nor K 2, 3 but every subgraph with 2n 4 3 (log n) 2 edges contains a C 4, (n > n O ) .
47 citations
TL;DR: It is shown that the minimum number of slopes needed to draw a complete graph of n vertices is n and it is proved that for a complete graphs to be drawn using only n slopes its vertices must form a convex polygon.
Abstract: In this paper we will study the problem of drawing graphs with a minimum number of slopes. We willshow that the minimum number of slopes needed to draw a complete graph of n vertices is n. We will also prove that for a complete graph of n vertices to be drawn using only n slopes its vertices must form a convex polygon. Finally, we will present an algorithm which checks whether a complete graph of n vertices can he drawn using only slopes from a given set of n slopes
TL;DR: It is shown that the Ramsey number of any graph with n vertices in which no two vertices of degree at least 3 are adjacent is at most 12n, which settles the problem of Burr and Erdos.
Abstract: It is shown that the Ramsey number of any graph with n vertices in which no two vertices of degree at least 3 are adjacent is at most 12n. In particular, the above estimate holds for the Ramsey number of any n-vertex subdivision of an arbitrary graph, provided each edge of the original graph is subdivided at least once. This settles a problem of Burr and Erdos.
TL;DR: In this paper, determinantal formulas using the unoriented Laplacian matrix MMt are used to count certain spanning substructures of G. These formulas may be viewed as generalizations of the matrix tree theorem.
TL;DR: The triangle-free graphs with neither induced path of six vertices nor induced cycle of six Vertices and the triangle- free graphs withoutinduced path ofSix vertices in terms of dominating subgraphs are characterized.
TL;DR: The construction of a minimum broadcast graph with 63 vertices with 162 edges is described and it is shown that the graph has 162 edges.
TL;DR: Thep-intersection graph of a collection of finite sets {Si}i=1n is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |Si∩Sj|≥p.
Abstract: Thep-intersection graph of a collection of finite sets {S i } i=1 n is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S i ∩S j |≥p. Thep-intersection number of a graphG, herein denoted θ p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK n,n andp≥2, then θ p (K n, n )≥(n 2+(2p−1)n)/p. Whenp=2, equality holds if and only ifK n has anorthogonal double covering, which is a collection ofn subgraphs ofK n , each withn−1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.
TL;DR: The vertex deletion problem for weighted directed acyclic graphs (WDAGs) is examined, finding the objective is to delete the fewest number of vertices so that the resulting WDAG has no path of length >/spl delta/.
Abstract: Examines the vertex deletion problem for weighted directed acyclic graphs (WDAGs). The objective is to delete the fewest number of vertices so that the resulting WDAG has no path of length >/spl delta/. Several simplified versions of this problem are shown to be NP-hard. However, the problem is solved in linear time when the WDAG is a rooted tree, and in quadratic time when the WDAG is a series-parallel graph. >
TL;DR: It is proved that a bipartite graph G has a {P3, P4, P5}-factor if and only if i(G − S − M) ≦ 2|S| + |M| for all S ⊆ V(G) and independent M⊆ E(G).
Abstract: A path on n vertices is denoted by Pn. For any graph H, the number of isolated vertices of H is denoted by i(H). Let G be a graph. A spanning subgraph F of G is called a {P3, P4, P5}-factor of G if every component of F is one of P3, P4, and P5. In this paper, we prove that a bipartite graph G has a {P3, P4, P5}-factor if and only if i(G − S − M) ≦ 2|S| + |M| for all S ⊆ V(G) and independent M ⊆ E(G). © 1994 Wiley Periodicals, Inc.
23 Jan 1994
TL;DR: In this article, a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths (up to constant factors) in the random graph Gn;m for all edge densities above the connectivity threshold is presented.
Abstract: Given a graph G =( V;E) with n vertices, m edges, and a family of pairs of vertices in V , we are interested in nding for each pair (ai;bi) a path connecting ai to bi such that the set of paths so found is edge disjoint. (For arbitrary graphs the problem isNP-complete, although it is inP if is xed.) We present a polynomial time randomized algorithm for nding the optimal number of edge disjoint paths (up to constant factors) in the random graph Gn;m for all edge densities above the connectivity threshold. (The graph is chosen rst; then an adversary chooses the pairs of endpoints.) Our results give the rst tight bounds for the edge-disjoint paths problem for any nontrivial class of graphs.
TL;DR: It is shown that a k -vertex l -edge I (3, m )-Steiner distance stable graph, m ⩾4, is k - vertex l-edge I(3,m +1)-Steinerdistance stable.
01 Jan 1994
TL;DR: In this article, an algorithm is described which constructs a long path containing a selected vertex x in a 2-connected undirected simple graph G. The algorithm uses crossovers of order kM, where M is a fixed constant, to build a longer and longer path.
Abstract: An algorithm is described which constructs a long path containing a selected vertex x in a graph G. In hamiltonian graphs, it often finds a hamilton cycle or path. The algorithm uses crossovers of order kM, where M is a fixed constant, to build a longer and longer path. The method is based on theoretical methods often used to prove graphs hamiltonian. 1. Crossovers Let G be a 2-connected undirected simple graph on n vertices. If u,v 2 V (G), then u ! v means that u is adjacent to v (and so also v ! u). The reader is referred to (4) for other graph-theoretic terminology. In particular, a trail in G is a walk in which vertices may be repeated, but not edges. Since G is simple, paths and trails may be represented as sequences of vertices. This uniquely defines their edge-sets. If (w0,w1,...,wm) represents a trail Q, we use Q to denote both the trail itself (ie, a subgraph of G), its sequence of vertices, as well as its set of edges. The usage should always be clear from the context. Let x be a vertex of G. We want to find a long path P in G containing x. Initially, P = (x), a path of length 0. We then extend the path P as follows. u := x; v := x while 9w ! u such that w 62 P do P := P + uw; u := w; end while 9w ! v such that w 62 P do P := P + vw; v := w; end
Proceedings Article•
01 Jan 1994
TL;DR: In this paper, the problem of moving a mobile robot to a designated vertex t using the smallest number of steps possible has been studied, with exact and approximate algorithms for several cases.
Abstract: We are given a connected. undirected graph G on n vertices. There is a mobile robot on one of the vertices; this vertex is labeled s. Each of several other vertices contains a single movable obsracle. The robot and the obstacles may only reside at vertices, although they may be moved across edges. A vertex may never contain more than one object (robot/obstacle). In one step, we may move either the robot or one of the obstacles from its current position to a vacant vertex adjacent to U. Our goal is to move the robot to a designated vertex t using the smallest number of steps possible. The problem is a simple abstraction of a robot motion planning problem. with the geometry replaced by the adjacencies in the graph. Wc point out its connections to robot motion planning. We study its complexity. giving exact and approximate algorithms for several cases.
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TL;DR: The authors prove that every graph of order n and minimum degree at least $\[3/(6+\sqrt{3})\]n$ is Hamiltonian if and only if it is path-tough.
Abstract: A graph G is called path-tough, if, for each nonempty set S of vertices, the graph G-S can be covered by at most |S| vertex disjoint paths. The authors prove that every graph of order n and minimum degree at least $\[3/(6+\sqrt{3})\]n$ is Hamiltonian if and only if it is path-tough. Similar results involving the degree sum of two or three independent vertices, respectively, are given. Moreover, it is shown that every path-tough graph without three independent vertices of degree 2 contains a 2-factor. The authors also consider complexity aspects and prove that the decision problem of whether a graph is path-tough is NP-complete.
TL;DR: It is shown that it is possible, for any n and k satisfying 2⩽ k ⩽n and k ≠3, to generate permutations of {1... n } so that successive permutations differ in k consecutive positions.
25 Aug 1994
TL;DR: This work proposes an O(λ2¦V¦+¦Γ¦log λ)+¦E¦) algorithm for (λ + 1)ECA-SV with Γ(V), where λ is the edge-connectivity of Γ (the cardinality of a minimum cut separating two vertices ofΓ).
Abstract: The k-edge-connectivity augmentation problem for a specified set of vertices (kECA-SV for short) is defined by “Given a graph G=(V, E) and a subset Γ\(\subseteq \)V, find a minimum set E′ of edges,each connecting distinct vertices of V, such that G′=(V, E ∪ E′) has at least k edge-disjoint paths between any pair of vertices in Γ”. We propose an O(λ2¦V¦(¦V¦+¦Γ¦log λ)+¦E¦) algorithm for (λ + 1)ECA-SV with Γ(V), where λ is the edge-connectivity of Γ (the cardinality of a minimum cut separating two vertices of Γ). Also mentioned is an O(¦V¦ log ¦V¦+¦E¦) algorithm for a special case where λ is equal to the edge-connectivity of G.
TL;DR: It is shown that determining whether Jordan-curve arrangement graphs are Hamiltonian is NP-complete.
TL;DR: If G is a 2-connected graph with n vertices and minimum degree d, then the vertices of G can be covered by less than n/d cycles, which settles a conjecture of Enomoto, Kaneko and Tuza for 2- connected graphs.
Abstract: “If G is a 2-connected graph with n vertices and minimum degree d, then the vertices of G can be covered by less than n/d cycles. This settles a conjecture of Enomoto, Kaneko and Tuza for 2-connected graphs.”
TL;DR: The largest size of a total matching and the smallest size of the maximum set of vertices and edges that cover every vertex and edge in a threshold graph are determined.
TL;DR: In this paper the iterated behavior of thek-line graph operator is investigated and it turns out that the behavior is quite different fork = 2 (the well-known line graph case),k = 3, and k≥4.
Abstract: For integerskÂ?2, thek-line graph Lk(G) of a graph G is defined as a graph whose vertices correspond to the complete subgraphs onk vertices in G with two distinct vertices adjacent if the corresponding complete subgraphs have 1 common vertices inG. We define iteratedk-line graphs byL k n (G) Â?L k (L k nÂ?1 (G), whereL k 0 (G) Â?G. In this paper the iterated behavior of thek-line graph operator is investigated. It turns out that the behavior is quite different fork = 2 (the well-known line graph case),k = 3, and kÂ?4.
TL;DR: In the sequel to the present paper, a criterion of the existence of edge-disjoint paths connecting certain vertices in a planar graph with three holes is given, provided that the so-called "parity condition" holds.