Assume that each vertex of a graph G is assigned a constant number q of nonnegative integer weights, and that q pairs of nonnegative integers l(i) and u(i), 1 <= i <= q, are given. One wishes to partition G into...
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Assume that each vertex of a graph G is assigned a constant number q of nonnegative integer weights, and that q pairs of nonnegative integers l(i) and u(i), 1 <= i <= q, are given. One wishes to partition G into connected components by deleting edges from G so that the total i-th weights of all vertices in each component is at least li and at most ui for each index i, 1 <= i <= q. The problem of finding such a "uniform" partition is NP-hard for series-parallel graphs, and is strongly NP-hard for general graphs even for q = 1. In this paper we show that the problem and many variants can be solved in pseudo-polynomial time for series-parallel graphs and partial k-trees, that is, graphs with bounded tree-width.
A vertex-ranking of a graph G is a labeling of the vertices of G with positive integers such that every path between two vertices with the same label i contains a vertex with label j > i. The minimum vertex-ranking...
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ISBN:
(纸本)9789843238146
A vertex-ranking of a graph G is a labeling of the vertices of G with positive integers such that every path between two vertices with the same label i contains a vertex with label j > i. The minimum vertex-ranking spanning tree problem is to find a spanning tree of a graph G whose vertex-ranking needs least number of labels. In this paper, we present:ani algorithm to solve the minimum vertex-ranking spanning tree problem on a series-parallel graph G in O(n(5) log(4) n) time, where n is the number of vertices in G.
This paper presents a bicriterion analysis of time/cost trade-offs for the single-machine scheduling problem where both job processing times and release dates are controllable by the allocation of a continuously nonre...
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This paper presents a bicriterion analysis of time/cost trade-offs for the single-machine scheduling problem where both job processing times and release dates are controllable by the allocation of a continuously nonrenewable resource. Using the bicriterion approach, we distinguish between our sequencing criterion, namely the makespan, and the cost criterion, the total resource consumed, in order to construct an efficient time/cost frontier. Although the computational complexity of the problem of constructing this frontier remains an open question, we show that the optimal job sequence is independent of the total resource being used;thereby we were able to reduce the problem to a sequencing one. We suggest an exact dynamic programming algorithm for solving small to medium sizes of the problem, while for large-scale problems we present some heuristic algorithms that turned out to be very efficient. Five different special cases that are solvable by using polynomial time algorithms are also presented. (c) 2005 Elsevier Ltd. All rights reserved.
Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total w...
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Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an "almost uniform" partition is called an (l, u)- partition. We deal with three problems to find an (l, u)- partition of a given graph;the minimum partition problem is to find an (l, u)- partition with the minimum number of components;the maximum partition problem is defined analogously;and the p-partition problem is to find an (l, u)- partition with a fixed number p of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time O(u(4)n) and the p-partition problem can be solved in time O( p(2)u(4)n) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width. (C) 2005 Elsevier B.V. All rights reserved.
We define a two-terminal directed acyclic graph (st-dag) characterized by a special structure of its mincuts and call it a nested graph. It is proved that every nested graph is series-parallel as well. We show that an...
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In this paper we give an algorithm to generates all series-parallel graphs with at most m edges. This algorithm generate each series-parallel graph in constant time on average.
In this paper we give an algorithm to generates all series-parallel graphs with at most m edges. This algorithm generate each series-parallel graph in constant time on average.
A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each ...
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A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L(v)| >= min{5, triangle + 1} for each vertex v and |L(e)| >= max{5, d(v) + 1, d(w) + 1} for each edge e = vw, where triangle is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with triangle + 1 colors if triangle >= 4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition. (C) 2004 Elsevier B. V. All rights reserved.
The total chromatic number of series-parallel graphs of maximum degree greater than or equal to 4 will be determined using the double inductions and the method of exchanging colors from the aspect of configuration pro...
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The total chromatic number of series-parallel graphs of maximum degree greater than or equal to 4 will be determined using the double inductions and the method of exchanging colors from the aspect of configuration property. Thus, the result of paper [7] is a special case of this paper.
The entire chromatic number χ_(vef) (G) of a plane graph G is the minimalnumber of colors needed for coloring vertices, edges and faces of G such that no two adjacent orincident elements are of the same color. Let G ...
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The entire chromatic number χ_(vef) (G) of a plane graph G is the minimalnumber of colors needed for coloring vertices, edges and faces of G such that no two adjacent orincident elements are of the same color. Let G be a series-parallel plane graph, that is, a planegraph which contains no subgraphs homeomorphic to K 4. It is proved in this paper that χ_(vef)(G)≤ max{8, Δ(G) + 2} and χ_(vef) (G) = Δ + 1 if G is 2-connected and Δ(G) ≥ 6.
In an orthogonal drawing of a planar graph G, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common...
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ISBN:
(纸本)3540309357
In an orthogonal drawing of a planar graph G, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of G is called an optimal orthogonal drawing if the number of bends is minimum among all orthogonal drawings of G. In this paper we give an algorithm to find an optimal orthogonal drawing of any given series-parallel graph of the maximum degree at most three. Our algorithm takes linear time, while the previously known best algorithm takes cubic time. Furthermore, our algorithm is much simpler than the previous one. We also obtain a best possible upper bound on the number of bends in an optimal drawing.
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