In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the doub...
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In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the double inductions and the method of exchanging colors from the aspect of configuration property. For series-parallel graphs of △(G) ≥ 5, △(G) ≤ x'as(G) ≤ △(G) + 1. Moreover, x'as(G) = △(G) + 1 if and only if it has two adjacent vertices of maximum degree, where △(G) and X'as(G) denote the maximum degree and the adjacent strong edge chromatic number of graph G respectively.
In project management, three quantities are often used by project managers: the earliest starting date, the latest starting date and the float of tasks. These quantities are computed by the Program Evaluation and Revi...
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In project management, three quantities are often used by project managers: the earliest starting date, the latest starting date and the float of tasks. These quantities are computed by the Program Evaluation and Review Techniques/Critical Path Method (PERT/CPM) algorithm. When task durations are ill known, as is often the case at the beginning of a project, they can be modeled by means of intervals, representing the possible values of these task durations. With such a representation, the earliest starting dates, the latest starting dates and the floats are also intervals. The purpose of this paper is to give efficient algorithms for their computation. After recalling the classical PERT/CPM problem, we present several properties of the concerned quantities in the interval-valued case, showing that the standard criticality analysis collapses. We propose an efficient algorithm based on path enumeration to compute optimal intervals for latest starting times and floats in the general case, and a simpler polynomial algorithm in the case of series-parallel activity networks.
Given an undirected network G = (V, E), a vector of nonnegative integers r = (r(nu) : nu is an element of V) associated with the nodes of G and weights on the edges of G, the survivable network design problem is to de...
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Given an undirected network G = (V, E), a vector of nonnegative integers r = (r(nu) : nu is an element of V) associated with the nodes of G and weights on the edges of G, the survivable network design problem is to determine a mininium-weight subnetwork of G such that between every two nodes u, v of V, there are at least min {r (u), r(nu)} edge-disjoint paths. In this paper we study the polytope associated with the solutions to that problem. We show that when the underlying network is series-parallel and r(nu) is even for all nu is an element of V, the polytope is completely described by the trivial constraints and the so-called cut constraints. As a consequence, we obtain a polynomial time algorithm for the survivable network design problem in that class of networks. This generalizes and unifies known results in the literature. We also obtain a linear description of the polyhedron associated with the problem in the same class of networks when the use of more than one copy of an edge is allowed. (c) 2004 Elsevier B.V. All rights reserved.
Suppose G is a series-parallel graph. It was proved in [3] that either chi(c)(G) = 3 or chi(c)(G) less than or equal to 8/3. So none of the rationals in the interval (8/3, 3) is the circular chromatic number of a seri...
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Suppose G is a series-parallel graph. It was proved in [3] that either chi(c)(G) = 3 or chi(c)(G) less than or equal to 8/3. So none of the rationals in the interval (8/3, 3) is the circular chromatic number of a series-parallel graph. This paper proves that for every rational r epsilon [2, 8/3] boolean OR {3} there exists a series-parallel graph G with chi(c)(G) = r. (C) 2004 Wiley Periodicals, Inc.
A feedback vertex set is a subset of vertices in a graph, whose deletion from the graph makes the resulting graph acyclic. In this paper, we study the minimum-weight feedback vertex set problem in series-parallel grap...
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A feedback vertex set is a subset of vertices in a graph, whose deletion from the graph makes the resulting graph acyclic. In this paper, we study the minimum-weight feedback vertex set problem in series-parallel graphs and present a linear-time exact algorithm to solve it.
Let G be a graph, and let each vertex v of G have a positive integer weight omega(v). A multicoloring of G is to assign each vertex v a set of omega(v) colors so that any pair of adjacent vertices receive disjoint set...
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Let G be a graph, and let each vertex v of G have a positive integer weight omega(v). A multicoloring of G is to assign each vertex v a set of omega(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given series-parallel graph G with the minimum number of colors in time 0(nW), where n is the number of vertices and W is the maximum weight of vertices in G.
This paper studies the following variations of arboricity of graphs. The vertex ( respectively, tree) arboricity of a graph G is the minimum number va( G) ( respectively, ta( G)) of subsets into which the vertices of ...
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This paper studies the following variations of arboricity of graphs. The vertex ( respectively, tree) arboricity of a graph G is the minimum number va( G) ( respectively, ta( G)) of subsets into which the vertices of G can be partitioned so that each subset induces a forest ( respectively, tree). This paper studies the vertex and the tree arboricities on various classes of graphs for exact values, algorithms, bounds, hamiltonicity and NP-completeness. The graphs investigated in this paper include block-cactus graphs, series-parallel graphs, cographs and planar graphs.
Assume that each edge e of a graph G is assigned a list (set) L(e) of colors. Then an edge-coloring of G is called an L-edge-coloring if each edge e of G is colored with a color contained in L(e). In this paper, we pr...
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Assume that each edge e of a graph G is assigned a list (set) L(e) of colors. Then an edge-coloring of G is called an L-edge-coloring if each edge e of G is colored with a color contained in L(e). In this paper, we prove that any series-parallel simple graph G has an L-edge-coloring if \L(e)\ greater than or equal to max {3,d(upsilon),d(omega)} for each edge e = upsilonw, where d(upsilon) and d(omega) are the degrees of the ends upsilon and omega of e, respectively. Our proof yields a linear algorithm for finding an L-edge-coloring of series-parallel graphs.
Assume that each edge e of a graph G is assigned a list (set) L(e) of colors. Then an edge-coloring of G is called an L-edge-coloring if each edge e of G is colored with a color contained in L(e). It is known that any...
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Assume that each edge e of a graph G is assigned a list (set) L(e) of colors. Then an edge-coloring of G is called an L-edge-coloring if each edge e of G is colored with a color contained in L(e). It is known that any series-parallel simple graph G has an L-edge-coloring if either (i) \L(e)\ greater than or equal to max {4,d(v),d(w)} for each edge e = vw or (ii) the maximum degree of G is at most three and \L(e)\ greater than or equal to 3 for each edge e, where d(v) and d(w) are the degrees of the ends v and w of e, respectively. In this paper we give a linear-time algorithm for finding such an L-edge-coloring of a series-parallel graph G.
We give a geometric representation of free De Morgan bisemigroups,free commutative De Morgan bisemigroups, and free De Morgan bisemilattices by using labeled graphs.
We give a geometric representation of free De Morgan bisemigroups,free commutative De Morgan bisemigroups, and free De Morgan bisemilattices by using labeled graphs.
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