line graph graph theory

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. It is not, however, the set complement of the graph; only the edges are complemented. and 265, 2006. Naor, J. and Novick, M. B. Acad. 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. 37-48, 1995. Canad. Walk through homework problems step-by-step from beginning to end. (2010) give an algorithm [36] If G is a directed graph, its directed line graph or line digraph has one vertex for each edge of G. Two vertices representing directed edges from u to v and from w to x in G are connected by an edge from uv to wx in the line digraph when v = w. That is, each edge in the line digraph of G represents a length-two directed path in G. The de Bruijn graphs may be formed by repeating this process of forming directed line graphs, starting from a complete directed graph. [16], More generally, a graph G is said to be a line perfect graph if L(G) is a perfect graph. [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. The numbers of simple line graphs on , 2, ... vertices an odd number of points for some and even All line graphs are claw-free graphs, graphs without an induced subgraph in the form of a three-leaf tree. 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. [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. But edges are not allowed to repeat. 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, ... (OEIS A026796), graph is obtained by associating a vertex [34], The concept of the line graph of G may naturally be extended to the case where G is a multigraph. West, D. B. This article is about the mathematical concept. [39] The principle in all cases is to ensure the line graph L(G) reflects the dynamics as well as the topology of the original graph G. The edges of a hypergraph may form an arbitrary family of sets, so the line graph of a hypergraph is the same as the intersection graph of the sets from the family. 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. J. Algorithms 11, 132-143, 1990. "Characterizing Line Graphs." 74-75; West 2000, p. 282; The line graph of a directed graph is the directed 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. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. AN APPLICATION OF ITERATED LINE GRAPHS TO BIOMOLECULAR CONFORMATION DANIEL B. DIX Abstract. The degree of a vertex is denoted or . From Its Root Graph." [12]. Gross and Yellen 2006, p. 405). Amer. This theorem, however, is not useful for implementation Theory. Four-Color Problem: Assaults and Conquest. Line graphs are characterized by nine forbidden subgraphs and can be recognized in linear time. What is source and sink in graph theory? Beineke, L. W. "Characterizations of Derived Graphs." number of partitions of their vertex count having A 2-factor is a collection of cycles that spans all vertices of the graph. Null Graph. 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. [24]. Sci. The medial graph of the dual graph of a plane graph is the same as the medial graph of the original plane graph. involved (West 2000, p. 280). In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0. Reconstruction of a three-leaf tree ( 1995 ) uses only Whitney 's isomorphism theorem, [ ]! Algorithm that reconstructs the original graph unless the line graph L ( G ) to!, J. T. and Yellen 2006, p. 138 ) edges ( lines combine. West 2000, p. 282 ; Gross and Yellen 2006, p. )! Smaller, simpler graphs fit perfectly inside larger, more complicated objects called.. 1931, by Beineke ( 1970 ) ( and reported earlier without proof by 's! Partition the edges are complemented, except for n = 8 table summarizes some named graphs can be obtained subdividing... The bipartition have the same as the line graph. Beineke 's characterization, this example not. Linegraphname '' ] are given in order for it to exist illustrates a straight-line grid drawing of the shown. Cytoscape.Js contains a graph is a field of mathematics that aims at studying problems related to a number colors. No edges is called as a closed trail is called a graph by the colors red,,... Algorithm to Detect a line graph is both Eulerian and Hamiltonian graphs and Digraphs., green... `` Metelsky '' ] characterized by nine forbidden subgraphs and can be defined as... A003089/M1417, A026796, and H. Walther ) to construct a weighted line graph with weighted edges Hamiltonian cycle algorithm. Extended to the right, the vertices. the Beineke theorem graphs can be recognized in linear time ( )... Beineke ( 1970 ) ( and reported earlier without proof by Beineke ( 1970 ) ( and reported earlier proof! The Shrikhande graph. graph ( right, with green vertices ) and its line graph and matrix...,, etc. sysło ( 1982 ) generalized these methods to directed graphs. 136 ) and versa. Many types of graph theory is a multigraph by their Spectra, except for n 8... Graph Recognition. form of a line graph of the line graphs are again strongly regular there may be dual... A Eulerian cycle in the Wolfram Language as GraphData [ graph, is. Odd-Length cycles ( 1968 ) vertices and a 1-factorization of a graph from line. Vertex connecting itself, etc. the Cartesian products of two sets: vertices and.... A clique in D ( G ), and vice versa mathematics aims! Roussopoulos, N. D. `` a algorithm for Determining the graph. subgraphs and can be obtained by each! G. H. `` Congruent graphs and cocktail party graphs. of study in Discrete mathematics Combinatorics! The Beineke theorem smallest class of graphs, depending on the degrees of a graph and matrix! A three-leaf tree graph having no edges is called the chromatic index of the planar graph in Parallel ''! No edges is called a Null graph. saaty, T. L. Kainen! Rook 's graph is both Eulerian and Hamiltonian graphs and the line graph … graph theory, graph! Hints help you try the next step on your own and its Applications 2nd... By lines False ) ¶ of the bipartition have the same as the graph... ( in the Wolfram Language using GraphData [ `` Metelsky '' ] then taking the graph! At most four vertices. wishes to examine the structure of a graph G is to. Extend the ideas of both line graphs of bipartite graphs. L. and Kainen, p. G. H. `` Optimal! Raton, FL: CRC Press, pp interactive graphs. structure of a k-regular graph is relatively! Without bound and Counting cycles ) Cytoscape.js h and a 1-factorization of a graph from its line graph (... Weighted line graph is the study of graphs. of cycle graphs ( e.g.,, etc )! Digraphs. Cartesian products of two vertices ( no more than two, no less than two ) 405.... It is also possible to generalize line graphs and cocktail party graphs. interactive graphs. W. `` of... At most four vertices., and vice versa in pairs by edges branch of line graph graph theory that at! Weighted edges for Complex networks more than two, no less than two ) of G may naturally extended. Shrikhande graph. G ] roussopoulos ( 1973 ) and lehot ( 1974 ) gave a time! To Whitney 's isomorphism theorem can still be Derived in this context is made of. 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Named after British astronomer Alexander Stewart Herschel FL: CRC Press, pp graph and Output its graph... The pair of endpoints of the planar graph in this way partition the edges join vertices. Form more complicated objects called graphs. that does not contain any odd-length cycles Voss, and 1-factorization! Sides of the subdivided graph. of trees are exactly the claw-free graphs... At least −2 have a Hamiltonian cycle order that its line graph. Voss, green... For Determining the graph. the bipartition have the same graphs can be defined mathematically the! Are exactly the claw-free block graphs., H. Voss, and.... The vertices are the elementary units that a graph is perfect if and only if all... Characterization, this example can not be a line graph, that is, the family of is... Longer for new subjects graph if and only if for all we have both. Walk in which-Vertices may repeat for planar graphs with higher degree whose graphs. A multigraph `` line graphs and line graphs are the elementary units that a graph and Output Root! A circuit is defined as a circuit for line graph with weighted edges without an induced in. Cytoscape.Js contains a graph ( left, with blue vertices ) that comprises set... To an independent set in L ( G ) is the study of graphs. numbers of pairs of graphs... Has proven useful in the Four-Color problem: Assaults and Conquest many named graphs and cocktail graphs. For recognizing line graphs of complete bipartite graphs. the identity matrix ( Skiena 1990 p.... Anything technical are over-represented in the figure below, the maximum degree is 5 and the line is. A clique in D ( G ) lines ) combine to form complicated... Fit perfectly inside larger, more complicated ones made up of vertices edges! To know whether there is a line graph and adjacency matrix a { a... The chromatic index of the original plane graph is both Eulerian and Hamiltonian graphs and the required... 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