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Clique graph representations of ptolemaic graphs

Terry A. McKee
2010 Discussiones Mathematicae Graph Theory  
The first of these, called a clique tree representation, is easily generated from the clique graph of G (the intersection graph of the maximal complete subgraphs of G).  ...  Thus, a ptolemaic graph G has two kinds of intersection graph representations: one from being chordal, and the other from being distance-hereditary.  ...  Ptolemaic graphs therefore have two kinds of tree representations: both a clique tree because of being chordal, and a CC tree because of being distancehereditary.  ... 
doi:10.7151/dmgt.1520 fatcat:laqtj6bp3rcpziv2zmw3dixhr4

Vector representations of graphs

T.D. Parsons, Tomaž Pisanski
1989 Discrete Mathematics  
Various types of vector representations of graphs are related to Hadamard matrices, families of sets with prescribed intersections, and other combinatorial problems.  ...  Let G be a simple graph with vertices 1,2, . . . , n.  ...  Here we shall use Hadamard representations to prove that an interesting family of graphs is n-universal.  ... 
doi:10.1016/0012-365x(89)90171-4 fatcat:dsy5zsml5fgidiylqpkge2qzpy

Geometric representations of graphs [chapter]

2010 Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization  
Consider an orthogonal representation i → v i ∈ R d of a graph G.  ...  The skeleton of a polytope, as a representation of itself, is section-connected; and so is the rubber-band representation of a planar graph.  ...  of the representing vectors.  ... 
doi:10.1090/fim/027/06 fatcat:hozttmgogzcv3epbax462axddy

Succinct representations of graphs

Hana Galperin, Avi Wigderson
1983 Information and Control  
Hong (1980) gives an algorithm with this upper bound for a certain class of succinctly representable graphs.  ...  The goals one would like to achieve by using a succinct representation are: (1) Reduce the amount of space required to store the graph. (2) Improve the complexity of certain graph algorithms.  ... 
doi:10.1016/s0019-9958(83)80004-7 fatcat:gw4uz4sy4zejnni4e6ebdhbtju

Representations of graphs modulon

Anthony B. Evans, Gerd H. Fricke, Carl C. Maneri, Terry A. McKee, Manley Perkel
1994 Journal of Graph Theory  
In particular, we relate the representation number of the disjoint union of complete graphs to the existence of complete families of mutually orthogonal Latin squares.  ...  The representation number of graph G is the smallest n representing G. We review known results and investigate representation numbers for several new classes.  ...  These tools will be used subsequently in determining representations and representation numbers of graphs and classes of graphs.  ... 
doi:10.1002/jgt.3190180805 fatcat:nvwjfxocdzbyziei6qokxnzbqy

Implicit representation of graphs

Sampath Kannan, Moni Naor, Steven Rudich
1988 Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88  
In the usual representations of an n-vertex graph, the names of the vertices (i.e. integers from 1 to n) betray nothing about the graph itself.  ...  For example, we can label planar graphs with structure coding labels of size < 4 log n, which implies the existence of a graph with n 4 vertices that contains all n-vertex planar graphs as vertex induced  ...  Furthermore from the representation of any graph it is easy to derive the representation of any of its vertex induced subgraphs.  ... 
doi:10.1145/62212.62244 dblp:conf/stoc/KannanNR88 fatcat:osg3ugvzvvhgbbmxq4mwv5be2u

Convex Representations of Graphs

W. T. Tutte
1960 Proceedings of the London Mathematical Society  
Consider a convex representation R of a simple graph G in a plane IT, R being determined by a 1-1 mapping/of V(G) onto a set of points of IT.  ...  We then call H a convex representation of G in E 2 .  ... 
doi:10.1112/plms/s3-10.1.304 fatcat:h2vb54d2mvdmhgbexk66xcxsd4

Orthogonal Representation of Graphs [article]

Alberto Solís-Encina, José Ramón Portillo
2015 arXiv   pre-print
Orthogonal Graph Representations are essential tools for testing existence of hidden variables in quantum theory.  ...  Conducting experiments quantum contextuality or the information capacity of a quantum system are closely related to the orthogonal representations.  ...  of the rest of the graph, so we reduce the dimension of the graph to n/2.  ... 
arXiv:1504.03662v2 fatcat:naveiodfl5bqtnqzde2jw3xorq

Geometric representations of graphs

Giuseppe Di Battista, Roberto Tamassia
1998 Computational geometry  
Geometric representations, or drawings, of graphs are widely used in information visualization applications, and their study has become increasingly popular in the last decade.  ...  Research on graph drawing is being actively conducted from a variety of perspectives, including discrete mathematics, computational geometry, human-computer interaction, and computer graphics.  ...  This special issue on Geometric Representations of Graphs contains five papers addressing a variety of graph drawing problems in two and three dimensions.  ... 
doi:10.1016/s0925-7721(97)00012-6 fatcat:ubv4vwlpibgizlvjcx2uurzhxi

Tree representations of graphs

Nancy Eaton, Zoltán Füredi, Alexandr V. Kostochka, Jozef Skokan
2007 European journal of combinatorics (Print)  
A graph is chordal if and only if it is the intersection graph of some family of subtrees of a tree. Applying "tolerance" allows larger families of graphs to be represented by subtrees.  ...  It is known that both [3, 3, 1] and [3, 3, 2] are equal to the family of chordal graphs. Furthermore, one can easily observe that every graph G belongs to [3, 3, t] for some t.  ...  For h = 4, we get a (3, 3, 589)-representation of K n,n for every 590 ≤ n ≤ 3150. Hence we can set n 0 = 589.  ... 
doi:10.1016/j.ejc.2006.04.002 fatcat:tmurppq4knebrcvqamwpkpbigm

A plane graph representation of triconnected graphs

Shunsuke Ota, Ehab Morsy, Hiroshi Nagamochi
2010 Theoretical Computer Science  
graph in the plane.  ...  Let G = (V , E) be a graph, S = {s 1 , s 2 , s 3 } be a set of three vertices of V , and assume that G has a directional S-embedding ψ.  ...  the boundary B of H; (b) a region graph of the graph given inFig. 1(a).  ... 
doi:10.1016/j.tcs.2010.08.029 fatcat:2lknzogsa5gflgrio525xaoiiu

Graph representations of finite Abelian groups

Bohdan Zelinka
1981 Czechoslovak Mathematical Journal  
If in the undirected graph representation of (5 each undirected edge xy is substituted by the pair of directed edges xy, yx, we obtain a directed graph representation of (6 c We shall study graph representations  ...  For this undirected graph representation of © we can construct a directed graph representation of © in the way described in the proof of Proposition.  ... 
doi:10.21136/cmj.1981.101725 fatcat:r2oqa5avnrc7dprgy7yon6m5ci

Accurate Learning of Graph Representations with Graph Multiset Pooling [article]

Jinheon Baek, Minki Kang, Sung Ju Hwang
2021 arXiv   pre-print
Yet, obtaining an accurate representation for a graph further requires a pooling function that maps a set of node representations into a compact form.  ...  A simple sum or average over all node representations considers all node features equally without consideration of their task relevance, and any structural dependencies among them.  ...  further encodes relationships between them to enhance the representation power of a graph.  ... 
arXiv:2102.11533v4 fatcat:qdqktuojbbbx7ofm6dgqzmevx4

Minimal Euclidean representations of graphs [article]

Aidan Roy
2009 arXiv   pre-print
In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix.  ...  We also give an exact formula for the Euclidean representation number using the main angles of the graph.  ...  This research is supported by NSERC, MITACS, PIMS, iCORE, and the University of Calgary Department of Mathematics and Statistics Postdoctoral Program.  ... 
arXiv:0812.3707v3 fatcat:usebhiv42fce3fpxe6uogxqzbe

A Characterization of Uniquely Representable Graphs [article]

Péter G. N. Szabó
2020 arXiv   pre-print
In this paper, we give a characterization of uniquely representable graphs by showing that they are exactly the block graphs.  ...  A connected graph G is uniquely representable if there exists a unique metric betweenness structure with underlying graph G. It was implied by previous works that trees are uniquely representable.  ...  Graphs that bound their representation from below and from above arise as the natural generalization of uniquely representable graphs.  ... 
arXiv:1708.01272v4 fatcat:mgaox6lufjeahblyhoc3kqjrxa
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