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### Graph Equations for Line Graphs, Jump Graphs, Middle Graphs, Splitting Graphs And Line Splitting Graphs

2010 Mapana Journal of Sciences
For a graph G, let G, L(G), J(G) S(G), L,(G) and M(G) denote Complement, Line graph, Jump graph, Splitting graph, Line splitting graph and Middle graph respectively.  ...  The equality symbol '=' stands for on isomorphism between two graphs.  ...  M(G)=L 5 (H) Beineke has shown in (2] that a graph G is a line graph if and only if G has none of the nine specified graphs f,, i = 1,2, .. . ,9 as an induced subgroph.  ...

### Interaction graphs: Graphings

Thomas Seiller
2017 Annals of Pure and Applied Logic
This third paper extends this approach by considering a generalization of graphs named graphings, which is in some way a geometric realization of a graph.  ...  The strength of our approach lies in the fact that we interpret proofs by simpler structures -graphs -than Girard's constructions, while generalizing the latter since they can be recovered as special cases  ...  We will here consider graphings as a generalization of graph or, more acutely, as a realization of a graph.  ...

### Graph equations for line graphs, total graphs, middle graphs and quasi-total graphs

1984 Discrete Mathematics
The authors are also thankful to the referee for suggesting the name 'quasi-total graphs'.  ...  Then we obtain a graph defined a graph M(G) as an intersection graph O(F) on the vertex-set V(G) of any graph G.  ...  The solution of L(G)-~P(H) Beineke [2] has shown that a graph G is a line graph if and only if G has none of the nine specified graphs as an induced subgraph.  ...

### Interaction Graphs: Graphings [article]

Thomas Seiller
2015 arXiv   pre-print
This third paper extends this approach by considering a generalization of graphs named graphings, which is in some way a geometric realization of a graph.  ...  The strength of our approach lies in the fact that we interpret proofs by simpler structures - graphs - than Girard's constructions, while generalizing the latter since they can be recovered as special  ...  We will here consider graphings as a generalisation of graph or, more acutely, as a realiser of a graph.  ...

### Equistable graphs, general partition graphs, triangle graphs, and graph products

Štefko Miklavič, Martin Milanič
2011 Discrete Applied Mathematics
graph is a general partition graph.  ...  In this paper we examine the connections between equistable graphs, general partition graphs and triangle graphs.  ...  : general partition graphs ⊆ strongly equistable graphs and equistable graphs ⊂ triangle graphs.  ...

### γ-graphs of graphs

Gerd H. Fricke, Sandra M. Hedetniemi, Stephen T. Hedetniemi, Kevin R. Hutson
2011 Discussiones Mathematicae Graph Theory
, say D 1 and In this paper we initiate the study of γ-graphs of graphs. 518 2.  ...  In this paper we consider the family of all γ-sets in a graph G and we define the γgraph G(γ) = (V (γ), E(γ)) of G to be the graph whose vertices V (γ) correspond 1-to-1 with the γ-sets of G, and two γ-sets  ...  Thus, the γ-graph of the graph G ′ is isomorphic to the tree T ′ .  ...

### Contracting Graphs to Split Graphs and Threshold Graphs [article]

Leizhen Cai, Chengwei Guo
2013 arXiv   pre-print
In these problems we are given a graph G and an integer k and asked whether G can be modified into a split graph or a threshold graph, respectively, by contracting at most k edges.  ...  We present an FPT algorithm for Split Contraction, and prove that Threshold Contraction on split graphs, i.e., contracting an input split graph to a threshold graph, is FPT when parameterized by the number  ...  Threshold graphs constitute a subclass of both split graphs and interval graphs in graph theory.  ...

### Graphs and graph algorithms [chapter]

2013 Scientific Programming
There is a wide range of computational tasks on graphs: Connectivity and components, Path-finding and traversals, including route finding, graph-searching, exhaustive cycles (Eulerian and Hamiltonian),  ...  Embedding problems, eg planarity -embedding in the plane, Matching ('marriage problems'), graph colouring and partitioning,  ...  There is a variety of different kinds of graphs: Undirected graphs, Directed graphs, Multi-graphs, Labelled graphs (either node-labelled, or edge-labelled, or both).  ...

### Drawing Graphs Within Graphs

Paul Holleis, Thomas Zimmermann, Daniel Gmach
2005 Journal of Graph Algorithms and Applications
This paper is based on a submission to the Graph Drawing Contest 2003 "Drawing Graphs within Graphs" [5] .  ...  In this paper, we present an approach for drawing graphs within graphs that first produces a layout for the subgraphs thus increasing their locality.  ...  Brandenburg, University of Passau, for the organization of the Graph Drawing Contest 2003 and for his constructive comments earlier versions of this paper.  ...

### Fractal graph and integral graph: C-type hyper graph in graph theory

Yaozhi Jiang
2019 Applied Mathematical Sciences
A new conception for  C type hyper graph is defined by fractal graph and integral graph in this paper.  C type hyper graph is unified the general graph and  C type hyper graph, i.e. any general graph  ...  can be seen a integral graph from a  C type hyper graph, and otherwise: any  C type hyper graph can be seen a fractal graph from a general graph.  ...  We call the graph on left is the integral graph of the fractal graph on right, and the graph on right is the fractal graph of the integral graph on left.  ...

### Graphs isomorphic to their path graphs

Martin Knor, Ľudovít Niepel
2002 Mathematica Bohemica
Path graphs were investigated by Broersma and Hoede in [2] as a natural generalization of line graphs, since P 1 (G) is the line graph L(G) of G (for further connections to line graphs see [6] ).  ...  Distance properties of path graphs are studied in [1] , [3] and [5] , and in [4] graphs with connected P 3 -path graphs are characterized.  ...

### Graph Coloring: Comparing Cluster Graphs to Factor Graphs [article]

Simon Streicher, Johan du Preez
2021 arXiv   pre-print
In contrast to the prevalent literature that uses factor graphs for this purpose, we instead approach it from a cluster graph perspective.  ...  Our experiments indicate a significant advantage for preferring cluster graphs over factor graphs, both in terms of accuracy as well as computational efficiency.  ...  the graph, and (d) a cluster graph configuration for this problem.  ...

### Small directed graphs as neighbourhood graphs

1988 Czechoslovak Mathematical Journal
Now consider the graphs without double edges. We shall determine the graphs G for particular graphs Я. If H is an empty graph, then G is any graph without edges.  ...  The graph G 1]L is in Fig. 4 , the graph G 12 is in Fig. 5 , the graph G 13 is in Fig. 6 . The graph G 14 is the complete directed graph with four vertices. Fig. 3 . 3 Fig. 3.  ...

### Graph equations for line graphs and total graphs

Dragǒs M. Cvetković, Slobodan K. Simić
1975 Discrete Mathematics
Hence a complete graph.  ...  r disconmxted, In each of the possibilities mentioned above, it is easy to find corresponding graphs G and so eq. (2) iz; solved.  ...

### Biclique Graphs of K_3-free Graphs and Bipartite Graphs [article]

Marina Groshaus, André Luiz Pires Guedes
2020 arXiv   pre-print
The biclique graph of a graph G, KB(G), defined as the intersection graph of the bicliques of G, was introduced and characterized in 2010.  ...  We prove that KB(G) is the square graph of a particular graph which we call Mutually Included Biclique Graph of G (KB_m(G)).  ...  The biclique graph of a graph G, denoted by KB(G), is the intersection graph of the bicliques of G.  ...
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