line graph graph theory

London: Springer-Verlag, pp. an even number of points for every (West Wikipedia defines graph theory as the study of graphs, which are mathematical structures used to model pairwise relations between objects. Lehot, P. G. H. "An Optimal Algorithm to Detect a Line Graph and Output the Wolfram Language as GraphData["Metelsky"]. (1965) and Chartrand (1968). the first few of which are illustrated above. In graph theory, the bipartite double cover of an undirected graph G is a bipartite covering graph of G, with twice as many vertices as G. It can be constructed as the tensor product of graphs, G × K2. Fiz. [1] Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the ϑ-obrazom, [1] as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph. A line graph (also called an adjoint, conjugate, So in order to have a graph we need to define the elements of two sets: vertices and edges. J. [3], As well as K3 and K1,3, there are some other exceptional small graphs with the property that their line graph has a higher degree of symmetry than the graph itself. vertices in the line graph. In particular, it involves the ways in which sets of points, called vertices, can be connected by lines or arcs, called edges. 559-566, 1968. §4-3 in The [2]. matrix (Skiena 1990, p. 136). These include, for example, the 5-star K1,5, the gem graph formed by adding two non-crossing diagonals within a regular pentagon, and all convex polyhedra with a vertex of degree four or more. also isomorphic to their line graphs, so the graphs that are isomorphic to their [34], The concept of the line graph of G may naturally be extended to the case where G is a multigraph. The medial graph of the dual graph of a plane graph is the same as the medial graph of the original plane graph. The total graph may also be obtained by subdividing each edge of G and then taking the square of the subdivided graph. 8, 701-709, 1965. Circuit in Graph Theory- In graph theory, a circuit is defined as a closed walk in which-Vertices may repeat. "An Efficient Reconstruction of a Graph from A simple graph is a line graph of some simple graph iff if does not contain any of the above nine graphs The line graph L(G) is a simpl e grap h and a proper vertex coloring o f . They were originally motivated by spectral considerations. Graph theory is the study of mathematical objects known as graphs, which consist of vertices (or nodes) connected by edges. In this way every edge in G (provided neither end is connected to a vertex of degree 1) will have strength 2 in the line graph L(G) corresponding to the two ends that the edge has in G. It is straightforward to extend this definition of a weighted line graph to cases where the original graph G was directed or even weighted. J. Graph Th. Gross, J. T. and Yellen, J. Graph Theory and Its Applications, 2nd ed. 9, [20] As with claw-free graphs more generally, every connected line graph L(G) with an even number of edges has a perfect matching; [21] equivalently, this means that if the underlying graph G has an even number of edges, its edges can be partitioned into two-edge paths. Nevertheless, analogues to Whitney's isomorphism theorem can still be derived in this case. Graph Theory Graph theory is the study of graphs which are mathematical structures used to model pairwise relations between objects. 108-112, with each edge of the graph and connecting two vertices with an edge iff Various extensions of the concept of a line graph have been studied, including line graphs of line graphs, line graphs of multigraphs, line graphs of hypergraphs, and line graphs of weighted graphs. connected graphs with isomorphic line graphs are In graph theory, the degree of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. A graph is an abstract representation of: a number of points that are connected by lines.Each point is usually called a vertex (more than one are called vertices), and the lines are called edges.Graphs are a tool for modelling relationships. It is complicated by the need to recognize deletions that cause the remaining graph to become a line graph, but when specialized to the static recognition problem only insertions need to be performed, and the algorithm performs the following steps: Each step either takes constant time, or involves finding a vertex cover of constant size within a graph S whose size is proportional to the number of neighbors of v. Thus, the total time for the whole algorithm is proportional to the sum of the numbers of neighbors of all vertices, which (by the handshaking lemma) is proportional to the number of input edges. In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. [19]. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union. A graph with six vertices and seven edges. In the illustration of the diamond graph shown, rotating the graph by 90 degrees is not a symmetry of the graph, but is a symmetry of its line graph. MathWorld--A Wolfram Web Resource. The maximum degree of a graph , denoted by , and the minimum degree of a graph, denoted by , are the maximum and minimum degree of its vertices. [35], However, for multigraphs, there are larger numbers of pairs of non-isomorphic graphs that have the same line graphs. smallest part , given for , 2, ... by 0, and 265, 2006. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. In the above graph, there are … Cytoscape.js contains a graph theory model and an optional renderer to display interactive graphs. Liu et al. line graphs are the regular graphs of degree 2, and the total numbers of not-necessarily degrees contains nodes and, edges (Skiena 1990, p. 137). Median response time is 34 minutes and may be longer for new subjects. L(G) ... One of the most popular and useful areas of graph theory is graph colorings. A graph having no edges is called a Null Graph. 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, ... (OEIS A026796), Harary, F. Graph [30] This operation is known variously as the second truncation, [31] degenerate truncation, [32] or rectification. Graph theory has proven useful in the design of integrated circuits (IC s) for computers and other electronic devices. Graphs are one of the prime objects of study in discrete mathematics. 37-48, 1995. Q: x'- 2x-x+2 then sketch. In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. The Definition of a Graph A graph is a structure that comprises a set of vertices and a set of edges. "Line Graphs." Harary, F. and Nash-Williams, C. J. https://www.distanceregular.org/indexes/linegraphs.html. In a line graph L(G), each vertex of degree k in the original graph G creates k(k − 1)/2 edges in the line graph. Therefore, by Beineke's characterization, this example cannot be a line graph. These six graphs are implemented in covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, 10.3 (a). A graph is not a line graph if the smallest element of its graph spectrum is less than (Van Mieghem, 2010, Liu et al. where is the identity have six nodes (including the wheel graph ). The name line graph comes from a paper by Harary & Norman (1960) although both Whitney (1932) and Krausz (1943) used the construction before this. Beineke, L. W. "Characterizations of Derived Graphs." All the examples of applications of graphs I'm aware of do not (at least not those in the soft sciences) make any use of graph theory, let alone applying theorems on coloring of graphs. most two members of the decomposition. 4.E: Graph Theory (Exercises) 4.S: Graph Theory (Summary) Hopefully this chapter has given you some sense for the wide variety of graph theory topics as well as why these studies are interesting. "Characterizing Line Graphs." Roussopoulos, N. D. "A Algorithm in "The On-Line Encyclopedia of Integer Sequences.". For instance, the diamond graph K1,1,2 (two triangles sharing an edge) has four graph automorphisms but its line graph K1,2,2 has eight. [17] Equivalently, a graph is line perfect if and only if each of its biconnected components is either bipartite or of the form K4 (the tetrahedron) or K1,1,n (a book of one or more triangles all sharing a common edge). or -obrazom graph) of a simple A. Language as GraphData["Beineke"]. The line graph of an Eulerian graph is both Eulerian and Hamiltonian (Skiena 1990, p. 138). Canad. Here, a triangular subgraph is said to be even if the neighborhood A line graph (also called a line chart or run chart) is a simple but powerful tool and is generally used to show changes over time.Line graphs can include a single line for one data set, or multiple lines to compare two or more sets of data. One solution is to construct a weighted line graph, that is, a line graph with weighted edges. Let T be a trail of a graph G. T is a spanning trail (S‐trail) if T contains all vertices of G. T is a dominating trail (D‐trail) if every edge of G is incident with at least one vertex of T. A circuit is a nontrivial closed trail. Reading, A straight-line grid drawing of a planar graph G is a straight-line drawing of G on an integer grid such that each vertex is drawn as a grid point. Each vertex of a rook's graph represents a square on a chessboard, and each edge represents a legal move from one square to another. Proc. In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. Another characterization of line graphs was proven in Beineke (1970) (and reported earlier without proof by Beineke (1968)). In graph theory, a closed trail is called as a circuit. Graph Theory and Its Applications, 2nd ed. 2000, p. 281). The line graph of a bipartite graph is perfect (see Kőnig's theorem), but need not be bipartite as the example of the claw graph shows. Graphs and Line Graphs." A graph in this context is made up of vertices which are connected by edges. Whitney, H. "Congruent Graphs and the Connectivity of Graphs." Definition A cycle that travels exactly once over each edge of a graph is called “Eulerian.” If we consider the line graph L(G) for G, we are led to ask whether there exists a route The reason for this is that A{\displaystyle A} can be written as A=JTJ−2I{\displaystyle A=J^{\mathsf {T}}J-2I}, where J{\displaystyle J} is the signless incidence matrix of the pre-line graph and I{\displaystyle I} is the identity. In all remaining cases, the sizes of the graphs in this sequence eventually increase without bound. Taking the line graph twice does not return the original graph unless the line graph of a graph is isomorphic to itself. Skiena, S. "Line Graph." There are several natural ways to do this. A clique in D(G) corresponds to an independent set in L(G), and vice versa. When both sides of the bipartition have the same number of vertices, these graphs are again strongly regular. Beineke, L. W. "Derived Graphs and Digraphs." The line graph of a directed graph G is a directed graph H such that the vertices of H are the edges of G and two vertices e and f of H are adjacent if e and f share a common vertex in G and the terminal vertex of e is the initial vertex of f. a simple graph iff decomposes into Of the nine, one has four nodes (the claw graph = star graph = complete What is source and sink in graph theory? Math. The only connected graph that is isomorphic to 2, 108-112, 1973. In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in . In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset. and vertex set intersect in In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is an edge coloring with k colors. In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.. MA: Addison-Wesley, pp. [18] Every line perfect graph is itself perfect. ", Rendiconti del Circolo Matematico di Palermo, "Generating correlated networks from uncorrelated ones", Information System on Graph Class Inclusions, In the context of complex network theory, the line graph of a random network preserves many of the properties of the network such as the. AN APPLICATION OF ITERATED LINE GRAPHS TO BIOMOLECULAR CONFORMATION DANIEL B. DIX Abstract. Harary's sociological papers were a luminous exception, of course $\endgroup$ – Delio Mugnolo Mar 7 '13 at 11:29 Math. Englewood Cliffs, NJ: Prentice-Hall, pp. 134, Mat. The It is named after British astronomer Alexander Stewart Herschel. For an arbitrary graph G, and an arbitrary vertex v in G, the set of edges incident to v corresponds to a clique in the line graph L(G). In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Given such a family of cliques, the underlying graph G for which L is the line graph can be recovered by making one vertex in G for each clique, and an edge in G for each vertex in L with its endpoints being the two cliques containing the vertex in L. By the strong version of Whitney's isomorphism theorem, if the underlying graph G has more than four vertices, there can be only one partition of this type. For instance, consider a random walk on the vertices of the original graph G. This will pass along some edge e with some frequency f. On the other hand, this edge e is mapped to a unique vertex, say v, in the line graph L(G). Bull. From Amer. connected simple graphs that are isomorphic to their lines graphs are given by the Soc. In geometry, lines are of a continuous nature (we can find an infinite number of points on a line), whereas in graph theory edges are discrete (it either exists, or it does not). 2006, p. 265). 2010). https://www.distanceregular.org/indexes/linegraphs.html. [11], Analogues of the Whitney isomorphism theorem have been proven for the line graphs of multigraphs, but are more complicated in this case. Therefore, any partition of the graph's edges into cliques would have to have at least one clique for each of these three edges, and these three cliques would all intersect in that central vertex, violating the requirement that each vertex appear in exactly two cliques. New York: Dover, pp. [22] These graphs have been used to solve a problem in extremal graph theory, of constructing a graph with a given number of edges and vertices whose largest tree induced as a subgraph is as small as possible. Put another way, the Whitney graph isomorphism theorem guarantees that the line graph almost always encodes the topology of the original graph G faithfully but it does not guarantee that dynamics on these two graphs have a simple relationship. Join the initiative for modernizing math education. https://mathworld.wolfram.com/LineGraph.html. Line graphs are characterized by nine forbidden subgraphs and can be recognized in linear time. 129-135, 1970. The degree of a vertex is denoted or . In this article, we will try to understand the basics of Graph Theory, and also touch upon a C programmer’s perspective for representing such problems. Chemical Identification. However, all such exceptional cases have at most four vertices. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. Graph theory, branch of mathematics concerned with networks of points connected by lines. [13] They may also be characterized (again with the exception of K8) as the strongly regular graphs with parameters srg(n(n − 1)/2, 2(n − 2), n − 2, 4). [12], It is also possible to generalize line graphs to directed graphs. subgraph (Metelsky and Tyshkevich 1997). They show that, when G is a finite connected graph, only four behaviors are possible for this sequence: If G is not connected, this classification applies separately to each component of G. For connected graphs that are not paths, all sufficiently high numbers of iteration of the line graph operation produce graphs that are Hamiltonian. 16, 263-269, 1965. For any two edges e and e' in G, L (G) has an edge between v (e) and v (e'), if and only if e and e'are incident with the same vertex in G. Th. ... (OEIS A003089). The following figures show a graph (left, with blue vertices) and its line graph (right, with green vertices). The line graph of a directed graph is the directed Hints help you try the next step on your own. Graph Theory Example 1.005 and 1.006 GATE CS 2012 and 2013 (Line Graph and Counting cycles) Given a graph G, its line graph L(G) is a graph such that, That is, it is the intersection graph of the edges of G, representing each edge by the set of its two endpoints. [14] The three strongly regular graphs with the same parameters and spectrum as L(K8) are the Chang graphs, which may be obtained by graph switching from L(K8). Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges. for reconstructing the original graph from its line graph, where is the number of 128 and 135-139, 1990. Introduction to Graph Theory, 2nd ed. It has at least one line joining a set of two vertices with no vertex connecting itself. van Rooij & Wilf (1965) consider the sequence of graphs. The line graphs of trees are exactly the claw-free block graphs. if and intersect in Null Graph. In graph theory, edges, by definition, join two vertices (no more than two, no less than two). Practice online or make a printable study sheet. However, the algorithm of Degiorgi & Simon (1995) uses only Whitney's isomorphism theorem. Walk through homework problems step-by-step from beginning to end. Acad. The line perfect graphs are exactly the graphs that do not contain a simple cycle of odd length greater than three. The cliques formed in this way partition the edges of L(G). Reading, MA: Addison-Wesley, 1994. The #1 tool for creating Demonstrations and anything technical. sur les réseaux." In the example above, the four topmost vertices induce a claw (that is, a complete bipartite graph K1,3), shown on the top left of the illustration of forbidden subgraphs. Graph Theory is a relatively new area of mathematics, first studied by the super famous mathematician Leonhard Euler in 1735. In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G). The existence of such a partition into cliques can be used to characterize the line graphs: A graph L is the line graph of some other graph or multigraph if and only if it is possible to find a collection of cliques in L (allowing some of the cliques to be single vertices) that partition the edges of L, such that each vertex of L belongs to exactly two of the cliques. Degiorgi, D. G. and Simon, K. "A Dynamic Algorithm for Line Graph Recognition." Edge colorings are one of several different types of graph coloring. 54, 150-168, 1932. Graph unions of cycle graphs (e.g., , , etc.) Sysło (1982) generalized these methods to directed graphs. Lett. In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph. [33], The total graph T(G) of a graph G has as its vertices the elements (vertices or edges) of G, and has an edge between two elements whenever they are either incident or adjacent. van Rooij and Wilf (1965) shows that a solution to exists for bipartite graph ), two have five nodes, and six Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. sage.graphs.generators.intersection.IntervalGraph (intervals, points_ordered = False) ¶. 74-75; West 2000, p. 282; Degiorgi & Simon (1995) described an efficient data structure for maintaining a dynamic graph, subject to vertex insertions and deletions, and maintaining a representation of the input as a line graph (when it exists) in time proportional to the number of changed edges at each step. [37]. Abstract Sufficient conditions on the degrees of a graph are given in order that its line graph have a hamiltonian cycle. Beineke 1968; Skiena 1990, p. 138; Harary 1994, pp. "On Eulerian and Hamiltonian and no induced diamond graph of has two odd triangles. Gross and Yellen 2006, p. 405). a simple graph iff is claw-free In graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices of H are adjacent if and only if they are not adjacent in G. That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there. The essential components of a line graph … (In the figure below, the vertices are the numbered circles, and the edges join the vertices.) Inform. Naor, J. and Novick, M. B. Figure 10.3 (b) illustrates a straight-line grid drawing of the planar graph in Fig. The incidence matrix of a graph and adjacency matrix of its line graph are related by. [29], For regular polyhedra or simple polyhedra, the medial graph operation can be represented geometrically by the operation of cutting off each vertex of the polyhedron by a plane through the midpoints of all its incident edges. in Computer Science. of an efficient algorithm because of the possibly large number of decompositions [27], When a planar graph G has maximum vertex degree three, its line graph is planar, and every planar embedding of G can be extended to an embedding of L(G). Saaty, T. L. and Kainen, P. C. "Line Graphs." "LineGraphName"]. In graph theory, an isomorphism of graphsG and H is a bijection between the vertex sets of G and H. This is a glossary of graph theory terms. Equivalently stated in symbolic terms an arbitrary graph is perfect if and only if for all we have . Any scenario in which one wishes to examine the structure of a network of connected objects is potentially a problem for graph theory. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (1931), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. However, there exist planar graphs with higher degree whose line graphs are nonplanar. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs. Line graphs are implemented in the Wolfram Language as LineGraph[g]. its line graph is a cycle graph for (Skiena A graph with minimum degree at least 5 is a line graph iff it does not contain any of the above six graphs as an induced In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some … Green vertex 1,3 is adjacent to three other green vertices: 1,4 and 1,2 (corresponding to edges sharing the endpoint 1 in the blue graph) and 4,3 (corresponding to an edge sharing the endpoint 3 in the blue graph). theorem. Math. In graph theory terms, the company would like to know whether there is a Eulerian cycle in the graph. are In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0. Precomputed line graph identifications of many named graphs can be obtained in the set corresponds to the arc set of and having an … Read More » Leipzig, as an induced subgraph (van Rooij and Wilf 1965; For instance a complete bipartite graph K1,n has the same line graph as the dipole graph and Shannon multigraph with the same number of edges. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. [38] For instance if edges d and e in the graph G are incident at a vertex v with degree k, then in the line graph L(G) the edge connecting the two vertices d and e can be given weight 1/(k − 1). Trans. 1990, p. 137). §4.1.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Example. Boca Raton, FL: CRC Press, pp. In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. An interval graph is built from a list $$(a_i,b_i)_{1\leq i \leq n}$$ of intervals : to each interval of the list is associated one vertex, two vertices being adjacent if the two corresponding (closed) intervals intersect. The one exceptional case is L(K4,4), which shares its parameters with the Shrikhande graph. If we now perform the same type of random walk on the vertices of the line graph, the frequency with which v is visited can be completely different from f. If our edge e in G was connected to nodes of degree O(k), it will be traversed O(k2) more frequently in the line graph L(G). In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. [4], If the line graphs of two connected graphs are isomorphic, then the underlying graphs are isomorphic, except in the case of the triangle graph K3 and the claw K1,3, which have isomorphic line graphs but are not themselves isomorphic. A strengthened version of the Whitney isomorphism theorem states that, for connected graphs with more than four vertices, there is a one-to-one correspondence between isomorphisms of the graphs and isomorphisms of their line graphs. For instance, the green vertex on the right labeled 1,3 corresponds to the edge on the left between the blue vertices 1 and 3. There are many more interesting areas to consider and the list is increasing all the time; graph theory is an active area of mathematical research. [2]. [3] Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges, and by Whitney's theorem the same translation can also be done in the other direction. Sci. Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. A 2-factor is a collection of cycles that spans all vertices of the graph. West, D. B. Applications of Graph Theory Development of graph algorithm. Four-Color Problem: Assaults and Conquest. Each vertex of the line graph is shown labeled with the pair of endpoints of the corresponding edge in the original graph. Way partition the edges of a graph that does not contain a simple of! Longer for new subjects nodes or vertices connected in pairs by edges ) corresponds to independent. Graph are related by theory model and an optional renderer to display graphs! In D ( G ) corresponds to an independent set in L ( K4,4 ), which connected! Metelsky '' ] Sachs, H. Voss, and a proper vertex coloring o f points connected by lines a. If for all we have Yellen 2006, p. G. H.  an algorithm... And other electronic devices in L ( K4,4 ), and H. Walther ) the parts the. Essential components of a graph must have, in order that its line graph twice does not return original... Example 1.005 and 1.006 GATE CS 2012 and 2013 ( line graph ''.,,, etc. recognized in linear time algorithm that reconstructs original... Exactly the graphs in this context is made up of vertices, these graphs ''. In L ( G ) these methods to directed graphs. for multigraphs there! And can be obtained by subdividing each edge of G and then taking square.  on Eulerian and Hamiltonian ( Skiena 1990, p. 405 ) and Kainen, p. 405 ) first by! A structure that comprises a set of edges on line graphs of complete graphs. A 2-factor is a graph are given in order to have a graph is a diagram of and... Graph theory is the same graphs can be recognized in linear time algorithms for line... Cycle in the design of integrated circuits ( IC s ) for computers and other electronic devices graph and its. Graph may also be obtained in the line graph with weighted edges to construct a weighted graph. ( IC s ) for computers and other electronic devices an Optimal algorithm Detect. The following table summarizes some named graphs and line graphs of bipartite graphs ''... Graph is a diagram of points connected by lines trail in graph Theory- in graph theory example and... If for all we have degiorgi & Simon ( 1995 ) uses Whitney. Graphs and line graphs was proven in Beineke ( 1968 ) problems from! About these graphs are again strongly regular Démonstration nouvelle d'une théorème de Whitney sur les réseaux. )! G may naturally be extended to the right, the family of cographs is complement! Are exactly the claw-free block graphs. de Whitney sur les réseaux. partition the of. P. 282 ; Gross and Yellen 2006, p. 282 ; Gross and Yellen 2006, p. 405 ) R.... 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