For positive integers p and q, let G(p,q) be a class of graphs such that vertical bar E(G)vertical bar = 2p. We obtain an upper bound for this sum that is linear in Delta(k-1). These graphs include the planar, 1-plana...
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For positive integers p and q, let G(p,q) be a class of graphs such that vertical bar E(G)vertical bar <= p vertical bar V(G)vertical bar - q for every G is an element of G(p,q). In this paper, we consider the sum of the kth powers of the degrees of the vertices of a graph G is an element of G(p,q) with Delta(G) >= 2p. We obtain an upper bound for this sum that is linear in Delta(k-1). These graphs include the planar, 1-planar, t-degenerate, outerplanar, and series-parallel graphs.
For a graph G=(V,E) and a color set C, let f:E -> C be an edge-coloring of G in which two adjacent edges may have the same color. Then, the graph G edge-colored by f is rainbow connected if every two vertices of G ...
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For a graph G=(V,E) and a color set C, let f:E -> C be an edge-coloring of G in which two adjacent edges may have the same color. Then, the graph G edge-colored by f is rainbow connected if every two vertices of G have a path in which all edges are assigned distinct colors. Chakraborty et al. defined the problem of determining whether the graph colored by a given edge-coloring is rainbow connected. Chen et al. introduced the vertex-coloring version of the problem as a variant, and we introduce the total-coloring version in this paper. We settle the precise computational complexities of all the three problems with regards to graph diameters, and also characterize these with regards to certain graph classes: cacti, outer planer and series-parallel graphs. We then give FPT algorithms for the three problems on general graphs when parameterized by the number of colors in C;our FPT algorithms imply that all the three problems can be solved in polynomial time for any graph with n vertices if |C|=O(logn).
This paper considers single-machine scheduling problems with deteriorating jobs, i.e., jobs whose processing times are an increasing function of their starting times. In addition, the jobs are related by a series-para...
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This paper considers single-machine scheduling problems with deteriorating jobs, i.e., jobs whose processing times are an increasing function of their starting times. In addition, the jobs are related by a series-parallel graph. It is shown that for the general linear problem to minimize the makespan, polynomial algorithms exist. It is also shown that for the proportional linear problem of minimization of the total weighted completion time, polynomial algorithms exist, too. (C) 2007 Elsevier Ltd. All rights reserved.
We examine variants of the critical node problem on specially structured graphs, which aim to identify a subset of nodes whose removal will maximally disconnect the graph. These problems lie at the intersection of net...
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We examine variants of the critical node problem on specially structured graphs, which aim to identify a subset of nodes whose removal will maximally disconnect the graph. These problems lie at the intersection of network interdiction and graph theory research and are relevant to several practical optimization problems. The two different connectivity metrics that we consider regard the number of maximal connected components (which we attempt to maximize) and the largest component size (which we attempt to minimize). We develop optimal polynomial-time dynamic programming algorithms for solving these problems on tree structures and on series-parallel graphs, corresponding to each graph-connectivity metric. We also extend our discussion by considering node deletion costs, node weights, and solving the problems on generalizations of tree structures. Finally, we demonstrate the computational efficacy of our approach on randomly generated graph instances. (c) 2011 Wiley Periodicals, Inc. NETWORKS, 2012
Let c(n) be themaximum number of cycles in an outerplanar graph with n vertices. We show that lim c(n)(1/n) exists and equals beta = 1.502837 ..., where beta is a constant related to the recurrence x(n+1) = 1 + x(n)(2...
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Let c(n) be themaximum number of cycles in an outerplanar graph with n vertices. We show that lim c(n)(1/n) exists and equals beta = 1.502837 ..., where beta is a constant related to the recurrence x(n+1) = 1 + x(n)(2), x(0) = 1. The same result holds for the larger class of series-parallel graphs.
We give a simple polynomial-time algorithm to exactly count the number of Euler tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the ...
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We give a simple polynomial-time algorithm to exactly count the number of Euler tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized series-parallel graphs includes all outerplanar graphs. We can perform the counting in time O (m Delta(3)), where Delta is the maximum degree of the graph with m edges. We use O (mn Delta(2) log Delta) bits to store intermediate values during our computations. To date, these are the first known polynomial-time algorithms to count or sample ETs of any class of graphs;there are no other known polynomial-time algorithms to even approximately count or sample ETs of any other class of graphs. The problem of counting ETs is known to be # P-complete for general graphs (Brightwell and Winkler, 2005 [2]) also for planar graphs (Creed, 2010 [3]). (C) 2011 Elsevier B.V. All rights reserved.
A proper edge coloring of G is r-acyclic if every cycle C contained in G is colored with at least min{vertical bar C vertical bar, r} colors. The r-acyclic chromatic index of a graph, denoted by a(r)'(G), is the m...
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A proper edge coloring of G is r-acyclic if every cycle C contained in G is colored with at least min{vertical bar C vertical bar, r} colors. The r-acyclic chromatic index of a graph, denoted by a(r)'(G), is the minimum number of colors required to produce an r-acyclic edge coloring. In this paper, we study 4-acyclic edge colorings by proving that a(4)' (G) <= 37 Delta(G) for every planar graph, a(4)' (G) <= max{2 Delta(G), 3 Delta(G) - 4} for every series-parallel graph and a(4)' (G) <= 2 Delta(G) for every outerplanar graph. In addition, we prove that every planar graph with maximum degree at least r and girth at least 5r + 1 has a(r)'(G) = Delta(G) for every r >= 4. (C) 2012 Elsevier B.V. All rights reserved.
Consider the NP-hard problem of, given a simple graph G, to find a series-parallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a 1/2-...
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Consider the NP-hard problem of, given a simple graph G, to find a series-parallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a 1/2-approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has n-1 edges and any series-parallel graph on n vertices has at most 2n-3 edges. We present a 7/12 -approximation for this problem and results showing the limits of our approach.
In a grid drawing of a planar graph, every vertex is located at a grid point, and every edge is drawn as a straight-line segment without any edge-intersection. It is known that every planar graph G of n vertices has a...
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In a grid drawing of a planar graph, every vertex is located at a grid point, and every edge is drawn as a straight-line segment without any edge-intersection. It is known that every planar graph G of n vertices has a grid drawing on an (n-2)x(n-2) or (4n/3)x(2n/3) integer grid. In this paper we show that if a planar graph G has a balanced partition then G has a grid drawing with small grid area. More precisely, if a separation pair bipartitions G into two edge-disjoint subgraphs G (1) and G (2), then G has a max {n (1),n (2)}xmax {n (1),n (2)} grid drawing, where n (1) and n (2) are the numbers of vertices in G (1) and G (2), respectively. In particular, we show that every series-parallel graph G has a (2n/3)x(2n/3) grid drawing and a grid drawing with area smaller than 0.3941n (2) (<(2/3)(2) n (2)).
In this paper, we study the minimum spanning tree problem with label selection, that is, the problem of finding a minimum spanning tree of a vertex-labeled graph where the weight of each edge may vary depending on the...
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In this paper, we study the minimum spanning tree problem with label selection, that is, the problem of finding a minimum spanning tree of a vertex-labeled graph where the weight of each edge may vary depending on the selection of labels of vertices at both ends. The problem is especially important as the application to mathematical OCR. It is shown that the problem is NP-hard. However, for the application to mathematical OCR, it is sufficient to deal with only graphs with small tree-width. In this paper, a linear-time algorithm for series-parallel graphs is presented. Since the minimum spanning tree problem with label selection is closely related to the generalized minimum spanning tree problem, their relation is discussed.
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