- Professor Emeritus of Tohoku University, Japan.edit
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. The total coloring problem is to find a total coloring of a given... more
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. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. However, no efficient algorithm has been known for the total coloring problem on partial k-trees although a polynomial-time algorithm of very high order has been known. In this paper, we give a linear-time algorithm for the total coloring problem on partial k-trees with bounded.
Several methods have been proposed for compressing the linkage data of a Web graph. Among them, the method proposed by Boldi and Vigna is known as the most efficient one. In the paper, we propose a new method to compress a Web graph. Our... more
Several methods have been proposed for compressing the linkage data of a Web graph. Among them, the method proposed by Boldi and Vigna is known as the most efficient one. In the paper, we propose a new method to compress a Web graph. Our method is more efficient than theirs with respect to the size of the compressed data. For example, our method needs only 1.99 bits per link to compress a Web graph containing 3,216,152 links connecting 325,557 pages, while the method of Boldi and Vigna needs 2.84 bits per link to compress the same Web graph.
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A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph play an important role in straight-line grid drawings, convex grid drawings, floor-plannings, graph encoding, etc. It is known that... more
A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph play an important role in straight-line grid drawings, convex grid drawings, floor-plannings, graph encoding, etc. It is known that the triconnectivity is a sufficient condition for their existence, but no necessary and sufficient condition has been known. In this paper, we present a necessary and sufficient condition for their existence, and show that a canonical decomposition, a realizer, a Schnyder labeling, an orderly spanning tree, and an outer triangular convex grid drawing are notions equivalent with each other. We also show that they can be found in linear time whenever a plane graph satisfies the condition.
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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... more
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)| ≥ max{3, d(v), d(w)} for each edge e = vw, where d(v) and d(w) are the degrees of the ends v and w of e, respectively. Our proof yields a linear algorithm for finding an L-edge-coloring of series-parallel graphs. key words: algorithm, list edge-coloring, series-parallel graph
あらまし 辺容量付き電力需給ネットワークはグラフ G で表現できる.G の各点は供給点あるいは需要点で あり,供給量あるいは需要量が割当てられており,各辺には辺容量が割当てられている.パラメトリックネット ワークでは,供給量,需要量及び辺容量は変数 λ の関数である.各需要点は,丁度一つの供給点からグラフ G の辺を通してその需要量だけの “電力” を受け取りたい.一方,各供給点は,幾つかの需要点へ G の辺を通して... more
あらまし 辺容量付き電力需給ネットワークはグラフ G で表現できる.G の各点は供給点あるいは需要点で あり,供給量あるいは需要量が割当てられており,各辺には辺容量が割当てられている.パラメトリックネット ワークでは,供給量,需要量及び辺容量は変数 λ の関数である.各需要点は,丁度一つの供給点からグラフ G の辺を通してその需要量だけの “電力” を受け取りたい.一方,各供給点は,幾つかの需要点へ G の辺を通して “電力”を送ることができるが,送る電力の合計はその供給量以下である.無論各辺を流れる電力フローはその辺 の容量以下でなければならない.このようなことが可能かどうか調べたい.また不可能ならば,全ての需要量を 一様に r(0 ≤ r < 1) 倍に減少させて,そのようなことを可能にしたい.このような r の最大値 r∗ を求めたい. 本論文では,木であるグラフ G に対してこれらの問題を解くアルゴリズムを与える. キーワード アルゴリズム,木,最大供給率問題,パラメトリックネットワーク,分割問題
A rectangle-of-influence drawing of a plane graph G is a straight-line planar drawing of G such that there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of any edge. In this paper, we show that... more
A rectangle-of-influence drawing of a plane graph G is a straight-line planar drawing of G such that there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of any edge. In this paper, we show that any 4-connected plane graph G with four or more vertices on the outer face has a rectangle-of-influence drawing in an integer grid such that W + H ≤ n, where n is the number of vertices in G, W is the width and H is the height of the grid. Thus the area W x H of the grid is at most [(n-1)/2] [(n-1)/2]. Our bounds on the grid sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose drawings need grids such that W + H = n - 1 and W x H = [(n-1)/2]. [(n-1)/2]. We also show that the drawing can be found in linear time.
A c-vertex-ranking of a graph G for a positive integer c is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices... more
A c-vertex-ranking of a graph G for a positive integer c is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. A c-vertex-ranking is optimal if the number of labels used is as small as possible. We present sequential and parallel algorithms to find an optimal c-vertex-ranking of a partial k-tree, that is, a graph of treewidth bounded by a fixed integer k. The sequential algorithm takes polynomial-time for any positive integer c. The parallel algorithm takes O(logn) parallel time using a polynomial number of processors on the common CRCW PRAM, where n is the number of vertices in G.
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MULTICOMMODITY FLOWS IN PLANAR NETWORKS 301 We now have the following theorem. Theorem 2. Algorithm MULTIFLÓW correctly finds multiconunodity flows of given demands in a pianar network N = {G,P,c) if all the sources and sinks are on the... more
MULTICOMMODITY FLOWS IN PLANAR NETWORKS 301 We now have the following theorem. Theorem 2. Algorithm MULTIFLÓW correctly finds multiconunodity flows of given demands in a pianar network N = {G,P,c) if all the sources and sinks are on the boundary of the ...
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SUMMARY Graph drawing addresses the problem of constructing ge-ometric representation of information and finds applications in almost ev-ery branch of science and technology. Efficient algorithms are essential for automatic drawings of... more
SUMMARY Graph drawing addresses the problem of constructing ge-ometric representation of information and finds applications in almost ev-ery branch of science and technology. Efficient algorithms are essential for automatic drawings of graphs, and hence a lot of research ...
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ABSTRACT
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A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex... more
A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.
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Spanning trees rooted at a vertex r of a graph G are independent if, for each vertex v in G, all the paths connecting v and r in the trees are pairwise internally disjoint. In this paper we give a linear-time algorithm to find the maximum... more
Spanning trees rooted at a vertex r of a graph G are independent if, for each vertex v in G, all the paths connecting v and r in the trees are pairwise internally disjoint. In this paper we give a linear-time algorithm to find the maximum number of independent spanning trees rooted at any given vertex r in partial k-trees
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Several routing problems such as VLSI river routing and single-layer routing can be formulated as a problem of finding a Steiner forest in a planar (grid) graph. Given an unweighted planar graph G together with nets of terminals, our... more
Several routing problems such as VLSI river routing and single-layer routing can be formulated as a problem of finding a Steiner forest in a planar (grid) graph. Given an unweighted planar graph G together with nets of terminals, our problem is to find a Steiner forest, i.e., vertex-disjoint trees, each of which interconnects all the terminals of a net. This paper gives an efficient algorithm to solve the problem for the case all terminals lie on two face boundaries of G. Also obtained is an algorithm for finding a maximum number of internally vertexdisjoint paths connecting two specified vertices in a planar graph G. Both run in 0( nlog n) time if G has n vertices.
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Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are integers such that 0 ≤ l ≤ u. One wishes to partition G into connected components by deleting edges from G so that the total weight of each... more
Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are integers such that 0 ≤ l ≤ u. 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