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.
An edge-coloring of a graph G is equitable if, for each vertex v of G, the number of edges of any one color incident with v differs from the number of edges of any other color incident with v by at most one. A graph G...
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ISBN:
(纸本)9783540725879
An edge-coloring of a graph G is equitable if, for each vertex v of G, the number of edges of any one color incident with v differs from the number of edges of any other color incident with v by at most one. A graph G is called equitable if G has an equitable edge-coloring with k colors for any integer k >= 1. A plane graph is series-parallel graph if it contains no subgraphs horneomorphic to K-4. In the paper, we prove that any simple and connected series-parallel graph is equitable if and only if it is not an odd circuit.
A set S of vertices in a graph G = (V, E) is an independent dominating set of G if no two vertices in S are adjacent and every vertex not in S is adjacent to a vertex in S. Suppose every vertex v is an element of V an...
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ISBN:
(纸本)9781614994848;9781614994831
A set S of vertices in a graph G = (V, E) is an independent dominating set of G if no two vertices in S are adjacent and every vertex not in S is adjacent to a vertex in S. Suppose every vertex v is an element of V and every edge e is an element of E are associated with a cost which is a real number, denoted by c(v) and c(e), respectively. The weighted independent domination problem is to find an independent dominating set D such that its total cost c(D) = Sigma(x is an element of D) c(x) + Sigma(x is not an element of D) min{c(x, y), for y is an element of D and (x, y) is an element of E} is minimum. In this paper, we propose a linear time algorithm for solving this problem in 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.
The minimum edge-ranking spanning tree (MERST) problem on a graph is to find a spanning tree of G whose edge-ranking needs least number of ranks. Although polynomial-time algorithm to solve the minimum edge-ranking sp...
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ISBN:
(纸本)9781424415502
The minimum edge-ranking spanning tree (MERST) problem on a graph is to find a spanning tree of G whose edge-ranking needs least number of ranks. Although polynomial-time algorithm to solve the minimum edge-ranking spanning tree problem on series-parallel graphs with bounded degrees has been found, but for the unbounded degrees no polynomial-time algorithm is known. In this paper, we prove that the minimum edge-ranking spanning tree problem on general series-parallel graph is NP-Complete.
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.
A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discover...
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A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discovered as a new standard for visualizing graphs. In this paper we study monotone drawings of series-parallel graphs in a variable embedding setting. We show that a series-parallel graph of n vertices has a straight-line planar monotone drawing on a grid of size O(n) x O(n(2)) and such a drawing can be found in linear time.
Let G be a series-parallel graph. In this paper, we present a linear algorithm of constructing an oriented binary decomposition tree of G. We use it to find 33 unavoidable subgraphs of G. Based on these 33 avoidable s...
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ISBN:
(纸本)9783540725879
Let G be a series-parallel graph. In this paper, we present a linear algorithm of constructing an oriented binary decomposition tree of G. We use it to find 33 unavoidable subgraphs of G. Based on these 33 avoidable subgraphs, we can determine the edge-face chromatic number, denoted by chi(ef) (G), of G where G is 2-connected and Delta(G) = 5. This completes the literature of determining chi(ef) (G) for 2-connected series-parallel graphs.
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.
Let c (n) be the maximum number of cycles in an outerplanar graph with n vertices. We show that lim c (n)1 / n exists and equals β = 1.502837 ..., where β is a constant related to the recurrence xn + 1 = 1 + xn2, x0...
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