Showing papers on "Split graph published in 1983"
TL;DR: Algorithms for two problems on intersection graphs of rectangles in the plane for finding the connected components of an intersection graph of n rectangles and an O(n log n) algorithm for finding a maximum clique of such a graph are described.
210 citations
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TL;DR: It is shown that a graph of large girth and minimum degree at least 3 share many properties with a graphs of large minimum degree, including a contraction containing a large complete graph.
101 citations
TL;DR: It is proved that a non-bipartite matching-covered graph contains K4 or K2⊕K3 (the triangular prism).
Abstract: We call a graphmatching-covered if every line belongs to a perfect matching. We study the technique of “ear-decompositions” of such graphs. We prove that a non-bipartite matching-covered graph containsK
4 orK
2⊕K
3 (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of τ-critical graphs.
96 citations
TL;DR: It is shown that for connected chordal graphs the center (the set of all vertices with minimum eccentricity) always induces a connected subgraph.
64 citations
01 Jan 1983
TL;DR: This paper surveys some of the progress made so far in covering and partitioning the edge set of a simple graph with a minimum number of complete subgraphs and presents a number of open problems.
Abstract: The problem of covering and partitioning the edge set of a simple graph with a minimum number of complete subgraphs has been studied by several writers over the years. This paper surveys some of the progress made so far and presents a number of open problems.
47 citations
TL;DR: The graph obtained by the removal of k disjoint edges from the complete graph on n vertices is shown to minimize all Schur-convex functions defined on the (Kirchoff) spectra of graphs, over all simple graphs on n Vertices and (n2)-k edges.
36 citations
TL;DR: A graph G is defined to be a unique eccentric point graph if each point of G has a unique maximum distance point and two construction procedures are given for generating families of u.e.p. p.
34 citations
01 Jan 1983
TL;DR: In this article, the authors considered only graphs without loops and multiple edges and the product of two vertex disjoint graphs G 1 and G 2 is the graph obtained by joining each vertex of G 1 to each node of G 2 given n and the sample graphs L 1,, L λ, and the maximum number of edges.
Abstract: In this paper we consider only graphs without loops and multiple edges The product of two vertex disjoint graphs G 1 and G 2 is the graph obtained by joining each vertex of G 1 to each vertex of G 2 Given n and the sample graphs L 1,, L λ, we shall consider those graphs on n vertices which contain no L i, as a subgraph and have maximum number of edges under this condition These graphs will be called extremal graphs for the L i’s
19 citations
TL;DR: It is shown that the following classes of graphs are recognizable (i.e. looking at the point-deleted subgraphs of a graph G one can decide whether G belongs to that class or not): (1) perfect graphs, (2) triangulated graph, (3) interval graphs,(4) comparability graphs, and (5) split graphs.
14 citations
TL;DR: In this paper, the authors generalize the construction of the group of automorphisms of a set, by constructing a group-graph associated with any given graph, i.e., the graph of reduced paths of the involutorial graph associated to the graph.
Abstract: In this paper we consider directed graphs with algebraic structures: group-graphs, ringgraphs, involutorial graphs, affine graphs, graphs of morphisms between graphs, graphs of reduced paths of an involutorial graph, etc. We show also how several well-known algebraic constructions can be carried over to graphs. As a typical example we generalize the construction of the group of automorphisms of a set, by constructing a group-graph associated with any given graphΓ. It is the group-graph of reduced paths of the involutorial graph associated to the graph of automorphisms ofΓ.
TL;DR: Some new properties of partitionable graphs are given which can be used to give a new proof that the Berge Strong Perfect Graph Conjecture holds for K"1","3-free graphs and it is shown that the SPGC also holds for the class of circle graphs.
TL;DR: In this paper, the complements of the 51 connected graphs in List E are connected graphs of order 7 through 9 with x = x = 3 and Corollary 2dimplies that there are many other connected graphs with x ≥ 3.
Abstract: is any connected graph of order 6 in List C.Of the 171 graphs which appear in these lists, 116 have x = x = 3. Inaddition to these the complements of the 51 graphs in List E areconnected graphs of order 7 through 9 with x = x = 3. And Corollary 2dimplies that there are many other graphs of order 7 through 9 with
01 Jan 1983
TL;DR: The clique covering number of a graph is the smallest number of complete subgraphs needed to cover its edge set as discussed by the authors, and it is defined as the smallest subgraph that can cover the edge set of a connected, cubic graph.
Abstract: The clique covering number of a graph is the smallest number of complete subgraphs needed to cover its edge-set. For each n, we determine the set of those integers which are clique covering numbers of connected, cubic graphs on n vertices. The analogous result for 4-regular graphs is stated.
01 Jan 1983
TL;DR: In this article, the authors present methods of compounding that result in families of graphs with large number of vertices for given values of the maximum degree and diameter D. From two graphs G1 and G2 on N 1 and N 2 vertices respectively, the compound graph G1[G2] on N1N2 vertices is obtained by connecting in some way N2 copies of G1.
Abstract: From two graphs G1 and G2 on N1 and N2 vertices respectively, the compound graph G1[G2] on N1N2 vertices is obtained by connecting in some way N2 copies of G1.
We present in this paper methods of compounding that result in families of graphs with large number of vertices for given values of the maximum degree ? and diameter D.
Patent•
20 Oct 1983
TL;DR: In this article, the authors proposed a method to enable a graph which is plain to see, by a method wherein a special printing is applied to a graph for input data exceeding the maximum printable value of data, and the graph is printed within the range of the maximum value.
Abstract: PURPOSE: To enable to form a graph which is plain to see, by a method wherein a special printing is applied to a graph for input data exceeding the maximum printable value of data, and the graph is printed within the range of the maximum value. CONSTITUTION: The maximum value in a graph is inputted, then a MAX key 12b is depressed, and a graph key 12a is depressed, whereby an XY plotter controller 15 reads the maximum value from a data buffer 14, writes the maximum value into an M-register in a RAM17, and writes the minimum value into an m-register. When M>m×10, the average of the data is calculated, is set on a MAX register, and when a maximum value is designated through a key input part 12, a scale of ordinate is printed in accordance with the maximum value. Then, the data exceeding the maximum value are printed in the form of split graph as shown, by a plotting part 20. Accordingly, even when the maximum data are quite different from other data, a graph in plain sight can be formed. COPYRIGHT: (C)1985,JPO&Japio
TL;DR: In this article, it was shown that a graph triangulating an orientable surface of genus γ = 1 + (ρ − 6)n/12 exists if and only if (ρ - 6) n ≡ 0 (mod 12).
Abstract: It is shown that, among regular graphs with n vertices of degree ρ, a graph triangulating an orientable surface of genus γ = 1 + (ρ – 6)n/12 exists if and only if (ρ – 6) n ≡ 0 (mod 12). A triangular imbedding for all such graphs is obtained with the help of the technique of flow graphs. Bibliography: 5 titles.